Discrete Artificial Boundary Conditions Vorgelegt Von Diplom

Discrete Artiﬁcial Boundary Conditions

vorgelegt von Diplom–Technomathematiker Matthias Ehrhardt aus Berlin

Vom Fachbereich Mathematik der Technischen Universitat¨ Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation

Promotionsausschuß: Vorsitzender: Prof. Dr. Udo Simon, TU Berlin Berichter: Prof. Dr. Anton Arnold, Universitat¨ des Saarlandes Berichter: Prof. Dr. Rolf Dieter Grigorieff, TU Berlin

Tag der wissenschaftlichen Aussprache: 25.5.2001

Berlin 2001 D 83

Contents

Acknowledgement iii Abstract v Zusammenfassung vii Introduction 1 Chapter 1. The Schrodinger¨ Equation 5 1. Transparent Boundary Conditions 5 2. Discrete Transparent Boundary Conditions 9 3. DTBC for non–compactly supported Initial Data 26 4. Numerical Inverse –Transformations 33 Chapter 2. The Convection–Diffusion Equation 41 1. Transparent Boundary Conditions 41 2. Discrete Transparent Boundary Conditions 44 3. Numerical Results 57 Chapter 3. The Wide–Angle Equation of Underwater Acoustics 67 1. Introduction to Underwater Acoustics 67 2. Transparent Boundary Conditions 72 3. Coupled Models for Underwater Acoustics 76 4. The Transparent Boundary Condition for an Elastic Bottom 78 5. Discrete Transparent Boundary Conditions 82 6. Numerical Examples 91 Conclusions and Perspectives 101

Appendix 103 The Laplace Transformation 103 The Inverse Laplace Transformation 104 The –Transformation 105 The Inverse –Transformation 106

Bibliography 109 Curriculum Vitae 115

i ii CONTENTS Acknowledgement

I want to express my deep gratitude to my Ph.D. supervisor Prof. Dr. Anton Arnold for his persevering support during the last years. I also thank Prof. Dr. Peter Markowich for offering me a position at the Technical University in Berlin. Special thanks to Prof. Dr. Juan Soler for inviting me as a TMR–Predoc to Granada. I am grateful to Prof. Lyness for answering all my questions about ENTCAF and Prof. Ron Mickens for many helpful comments on difference equations. I would like to thank Prof. Makrakis for inviting me to a workshop on underwater acoustics on Crete and Prof. Finn Jensen for his advice on test cases for underwater acoustics. I thank Prof. Mireille Levy for a fruitful email communication, Prof. John Papadakis and Dr. Frank Schmidt for helpful discussions. This work was ﬁnancially supported by the German Research Council (DFG) under Grant No. MA 1662/1–3 and MA 1662/2–2. The last sentence is dedicated to my parents who made it possible for me to study mathematics.

iii iv ACKNOWLEDGEMENT Abstract

When computing numerically the solution of a partial differential equation in an unbounded domain usually artificial boundaries are introduced to limit the computational domain. Special boundary conditions are derived at this artificial boundaries to approximate the exact whole– space solution. If the solution of the problem on the bounded domain is equal to the whole– space solution (restricted to the computational domain) these boundary conditions are called transparent boundary conditions (TBCs). This dissertation is concerned with transparent boundary conditions for convection–diffusion equations and general Schrodinger¨ –type pseudo–differential equations arising from “parabolic” equation (PE) models which have been widely used for one–way wave propagation problems in various application areas, e.g. seismology, optics and plasma physics. As a special case the Schrodinger¨ equation of quantum mechanics is included. Existing discretizations of these TBCs induce numerical reflections at this artificial boundary and also may destroy the stability of the used finite difference method. To overcome both problems we propose a new discrete TBC which is derived from the fully discretized whole– space problem. This discrete TBC is reflection–free and conserves the stability properties of the whole–space scheme. While we shall assume a uniform discretization in time, the interior spatial discretization may be nonuniform. The superiority of the new discrete TBC over existing discretizations is illustrated on several benchmark problems.

v vi ABSTRACT Zusammenfassung

Bei der numerischen Berechnung der Losung¨ einer partiellen Differentialgleichung auf einem unbeschrankten¨ Gebiet werden gewohnlich¨ kunstliche¨ Rander¨ eingefuhrt,¨ um das Rechen- gebiet zu beschrank¨ en. Spezielle Randbedingungen werden an diesen kunstlichen¨ Randern¨ hergeleitet, um die exakte Ganzraumlosung¨ zu approximieren. Falls die Losung¨ des Problems auf dem beschrankten¨ Gebiet mit der Ganzraumlosung¨ (eingeschrankt¨ auf das Rechengebiet) identisch ist, werden diese Randbedingungen als transparente Randbedingungen (TRB) beze- ichnet. Die vorliegende Doktorarbeit befaßt sich mit transparenten Randbedingungen fur¨ Konvek- tions–Diffusionsgleichungen und allgemeine Pseudodifferentialgleichungen vom Schrodingertyp.¨ Diese sogenannten “Parabolischen” Gleichungen finden weit verbreitete An- wendung bei 1–Weg–Wellenausbreitungsproblemen in vielen Bereichen, z.B. Seismologie, Op- tik und Plasmaphysik. Als Spezialfall ist die Schrodinger¨ gleichung der Quantenmechanik en- thalten. Existierende Diskretisierungen dieser TRB fuhren¨ zu numerischen Reflektionen an den kunstlichen¨ Randern¨ und zerstoren¨ haufig¨ die Stabilitat¨ der zugrundeliegenden finite Differen- zen Methode. Um beide Probleme zu losen,¨ fuhren¨ wir eine neue diskrete TRB ein, die direkt vom diskretisierten Ganzraumproblem hergeleitet wird. Diese diskrete TRB ist reflektions- frei und erhalt¨ die Stabilitatseigenschaften¨ des Ganzraumschemas. Wahrend¨ wir eine uniforme Diskretisierung in der Zeit voraussetzen mussen,¨ kann die innere Diskretisierung nichtuniform im Ort sein. Die Uberle¨ genheit der neuen diskreten TRB gegenuber¨ anderen existierenden Diskretisierungen von TRBen wird anhand von mehreren Beispielen illustriert.

vii viii ZUSAMMENFASSUNG Introduction

Many physical problems are described mathematically by a partial differential equation (PDE) which is defined on an unbounded domain. If one wants to solve such whole–space evolution problems numerically, one first has to make it finite dimensional. The standard approach is to restrict the computational domain by introducing artificial boundary conditions or absorbing layers. Further possible methods that can be applied are boundary element methods (BEM) (cf. [U17]) or infinite element methods (IEM) (see for example [U14]). In this dissertation we focus on the approach of artificial boundary conditions. If the initial data is supported on a finite domain Ω, one can construct boundary conditions (BCs) on ∂Ω with the objective to approximate the exact solution of the whole–space problem, restricted to Ω. Such BCs are called absorbing boundary conditions (ABCs) if they yield a well–posed (initial) boundary value problem (IBVP), where some “energy” is absorbed at the boundary. If the approximate solution coincides on Ω with the exact solution, one refers to these BCs as transparent boundary conditions (TBCs). Of course, these boundary conditions should lead to a well–posed (initial) boundary value problem. Additionally, it is desirable that the BCs are local in space and/or time to allow for an efficient numerical implementation. Here we are concerned with the construction and discretization of TBCs for general Schro-¨

dinger–type pseudo–differential evolution equations in 1D of the form

¤ ¤

¦§¦ ∆ ¥

p0 £ p1 V x t

¡ ¡ ¤

ψ ¤ ψ ¥ ¨ ¥ © ¥

i t ¢ 1 x IR t 0

¦§¦ ∆ ¥

(GS) 1 £ q1 V x t ¤

¤ I

¦ ¦ ¥ ψ ¥ ψ

x 0 ¢ x

where the real coefficients p0, p1, q1 are constant, ∆ denotes the Laplacian and the complex ¨ valued “potential” V is assumed to be given. As a special case (q1 ¢ 0, V IR) the Schrodinger¨ equation of quantum mechanics is included. Equations of the form (GS) arise from “parabolic” equation (PE) models, which have been widely used for 1–way wave propagation problems in various application areas, e.g. seismology [U4], [U5], optics and plasma physics (cf. the references in [U3]). In underwater acoustics they appear as wide angle approximation to the Helmholtz equation in cylindrical coordinates and are called wide angle parabolic equations (WAPE) [U18]. The usual strategy of employing TBCs for solving a whole–space problem numerically consists of first deriving an analytic TBC at the artificial boundary. These TBCs are typically nonlocal in time (of “memory–type”) and can be approximated by a local–in–time BC. Finally, the continuous BC must be discretized to use it with an interior discretization of the PDE. The analytic TBC for the Schrodinger¨ equation was independently derived by several authors from various application fields [T1], [T6], [T12], [T16], [T18], [T20]. While continuous TBCs fully solve the problem of cutting off the spatial domain for the analytical equation, their numerical discretization is far from trivial. Up to now the standard strategy is to derive first the TBC for the analytic equation, then to discretize it, and to use it 1 2 INTRODUCTION in connection with some appropriate numerical scheme for the PDE. The defect of this usual approach of discretizing continuous TBCs (discretized TBCs) is that the inner discretization of the PDE often does not “match” the discretization of the TBCs. There are two major problems of these existing consistent discretizations of the continuous TBC:

P1: The discretized TBCs for the Schrodinger¨ –type equation (GS) often destroy the stability of the whole–space ﬁnite difference scheme. Especially for the Schrodinger¨ equation of quantum mechanics the unconditional stability of the underlying Crank–Nicolson scheme is destroyed [T16] and the overall numerical scheme is rendered only conditionally stable [T6], [T16], [T19], [T26]. P2: The available discretizations often suffer from reduced accuracy (in comparison to the discretized whole–space problem) and induce numerical reﬂections at the boundary, particularly when using coarse grids.

In this dissertation we discuss in detail a recently developed approach (first outlined in [T1]) which overcomes both the stability problem (P1) and the problem of reduced accuracy (P2). In our discrete approach we propose to change the order of the two steps of the usual strategy, i.e. we first consider the discretization of the PDE on the whole space and then derive the TBC for the difference scheme directly on a purely discrete level. There are several advantages of the discrete approach. It completely avoids any numerical reflections at the boundary: no additional discretization errors due to the boundary conditions occur. The discrete TBC is already adapted to the inner scheme and therefore the numerical stability is often better–behaved than for a discretized differential TBC. An additional motivation for this discrete approach arises from the fact that the numerical scheme often needs more boundary conditions than the analytical problem can provide (especially hyperbolic equations, systems of equations and high–order schemes). In the literature the discrete approach did not gain much attention yet. The first discrete derivation of artificial boundary conditions was presented in [D2, Section 5]. This discrete approach was also used in [D5], [D6], [D7] for linear hyperbolic systems and in [D3] for the wave equation in one dimension, also with error estimates for the reflected part. In [D6] a discrete (nonlocal) solution operator for general difference schemes (strictly hyperbolic systems, with constant coefficients in 1D) is constructed. Lill generalized in [D4] the approach of Engquist and Majda [D2] to boundary conditions for a convection–diffusion equation and drops the standard assumption that the initial data is compactly supported inside the computational domain. However, the derived –transformed boundary conditions were approximated in order to get local–in–time artificial boundary conditions after the inverse –transformation. Here we construct discrete transparent boundary conditions (DTBC) for a Crank–Nicolson finite difference discretization of (GS) such that the overall scheme is unconditionally stable and as accurate as the discretized whole–space problem. The resulting DTBC is a generalization of the DTBC for the Schrodinger¨ equation in [T1]. The same strategy applies to the θ–scheme for convection–diffusion equations [P2] and was also used in [T10] for the wave equation in the frequency domain. Although this work concentrates on the discrete derivation of BCs, we will also consider the continuous problem, since the basic ideas of the construction and derivation carry over to the discrete case, e.g. we can use discrete versions of the L2–estimates. Moreover the well– posedness of the continuous problem is necessary for the stability of the numerical scheme. Just like the analytic TBC, the discrete TBC will be nonlocal in the time variable. INTRODUCTION 3 The dissertation is organized as follows: In Chapter 1 we introduce our new approach of deriving DTBCs: we first review the continuous TBC for the Schrodinger¨ equation in one space dimension and derive and analyze the discrete TBCs which are in the form of a discrete convolution. In order to obtain an efficient implementation one can easily localize these nonlocal in time DTBCs just by cutting off the rapidly decaying sequence of the convolution coefficients. Various numerical examples in Section 2 illustrate the superiority of our DTBC over existing discretizations. At the end of the first Chapter we discuss the DTBC in the case that

the initial data is not supported in the computational domain and show how to apply a numerical inverse –transformation if the exact inverse –transformation cannot be determined analytically. Chapter 2 shows how the presented method can be applied to a linear convection– diffusion equations and to a more general ﬁnite difference scheme. Finally, the third Chapter gives a more practical application of the DTBCs to underwater sound propagation. We discuss the DTBCs for the Crank–Nicolson scheme for the WAPE (which is of the form (GS)). In the conclusion we summarize the obtained results and discuss topics for further research. 4 INTRODUCTION CHAPTER 1

The Schrodinger¨ Equation

In this Chapter we want to clarify the approach of deriving discrete transparent boundary conditions. For simplicity we consider the time–dependent Schrodinger¨ equation in one space dimension. This approach was introduced by Arnold in [T1].

1. Transparent Boundary Conditions In this Section we shall sketch the derivation of the TBC and discuss the well–posedness of the resulting IBVP. Here we will treat the case of the Schroding¨ er equation

¦ ¥ ¨ ¥ © ¥

ψ ∆ψ ¥ ψ £ i ¢ V x t x IR t 0

(1.1.1) t 2 ¤

¤ I

¦ ¦ ¥ ψ ¥ ψ

x 0 ¢ x

¤ ¤ ¤

ψI 2 ¤ ∞

¦ ¥ ¦¨ ¦ ¥ ¦ where ¨ L IR , V t L IR and V x is piecewise continuous.

1.1. Derivation of the TBC. Our goal is to design transparent boundary conditions (TBCs) ¢

at x ¢ 0 and x L, such that the resulting IBVP is well–posed and its solution coincides with

¤ ¦ the solution of the whole–space problem restricted to 0 ¥ L . We make the following two basic assumptions:

ψI

A1: The initial data is supported in the computational domain 0 x L.

¥ ¦

A2: The given electrostatic potential is constant outside this ﬁnite domain: V x t ¢ 0 for

¥ ¦

x 0, V x t ¢ VL for x L.

REMARK 1.1. Without the ﬁrst assumption information would be lost, and the whole–space

¥ evolution could not be reproduced on the ﬁnite interval 0 L . The second assumption allows to explicitly solve the equation in the exterior of the computational domain (by the Laplace– method) which is the basic idea of the derivation of the TBC. In Section 3 we will see how to drop the assumption (A1). We present a formal derivation of the TBC for smooth (i.e. C1) solutions. Afterwards, the obtained TBC can be regarded for less regular solutions. The ﬁrst step is to cut the original

whole–space problem (1.1.1) into three subproblems, the interior problem on the domain 0 ψ ψ

x L, and a left and right exterior problem. They are coupled by the assumption that , x are ¢ continuous across the artiﬁcial boundaries at x ¢ 0, x L. The interior problem reads

2 ¤

ψ ¡ ∆ψ ψ ¥ ¦ ¥ ¥ © ¥ £ i ¢ V x t 0 x L t 0

t 2 ¤

¤ I

ψ ¥ ¦ ψ ¦ ¥

(1.1.2) x 0 ¢ x

¤ ¤ ¤

¦ ¦ ¥ ¦ ¥ ψ ¥ ψ

x 0 t ¢ T0 0 t

¤ ¤ ¤

¦ ¦ ¥ ¦ ψ ¥ ψ x L t ¢ TL L t

5 6 1. THE SCHRODINGER¨ EQUATION

T0 L denote the Dirichlet–to–Neumann maps at the boundaries, and they are obtained by solving the two exterior problems:

t 2 L

¥ ¦ ¥

v x 0 ¢ 0

¤ ¤

(1.1.3) ¤ Φ Φ

¥ ¦ ¦ ¥ © ¥ ¦ ¥ ¢ v L t ¢ t t 0 0 0

¤ ∞ ¥ ¦ ¥

v t ¢ 0

¤ ¤ ¤

Φ ¦ ¦ ¥ ¦ ¥

TL t ¢ vx L t and analogously for T0. Since the potential is constant in the exterior problems, we can solve

them explicitly by the Laplace method and thus obtain the two boundary operators T0 L needed in (1.1.2). This idea is illustrated in Figure 1.1.

ψ left exterior problem interior problem

(explicitly solvable) right ψ (x,t) exterior

output: vx (0,t) problem

input: I boundary data ψ (0,t) ψ

x 0 L

FIGURE 1.1. Schrodinger¨ equation : Construction idea for transparent boundary conditions

The Laplace transformation of v is given by

∞ ¤

¤ st

¥ ¦ ¥ ¦ ¥

(1.1.4) vˆ x s ¢ v x t e dt

η ξ ξ ¨ η £ where we set s ¢ i , IR, and 0 is ﬁxed, with the idea to later perform the limit η 0. Now the right exterior problem (1.1.3) is transformed to

2 VL

¥ © ¥

£ ¢ vˆxx £ i s i vˆ 0 x L

(1.1.5)

¤ Φ

¦ ¦ ¥ ˆ

vˆ L s ¢ s ¤

∞ ψ ¤ 2 ¥ ¦¨ ¦ Since its solutions have to decrease as x (since we have t L IR ), we obtain

¤ 2

i s i x L

ˆ ¦ ¦ ¥ Φ

(1.1.6) vˆ x s ¢ e s 1. TRANSPARENT BOUNDARY CONDITIONS 7

Hence the Laplace–transformed Dirichlet–to–Neumann operator TL reads

¤ ¤ ¤ π 2 VL

¡ i 4 ˆ ¥ ¦ ¦ ¥

Φ ¦ Φ

¢ ¢ (1.1.7) T s vˆ L s e s £ i s

L x ! and T0 is calculated analogously. Here, denotes the branch of the square root with nonnegative real part.

An inverse Laplace transformation yields the right TBC at x ¢ L:

¤ V

t i L τ

¤ π V

¦ ψ ¥ τ

2 L d L e

¡ i i t

ψ 4 ! τ ¥ ¦

(1.1.8) x L t ¢ e e d π dt 0 t ¡ τ

Similarly, the left TBC at x ¢ 0 is obtained as

t ¤

¤ π

¦ ψ ¥ τ 2 i d 0

ψ 4 ! τ ¥ ¦

(1.1.9) x 0 t ¢ e d π dt 0 t ¡ τ These BCs are nonlocal in t and of memory–type, thus requiring the storage of all previous time levels at the boundary in a numerical discretization. A second difﬁculty in numerically implementing (1.1.8), (1.1.9) is the discretization of the singular convolution kernel. A simple calculation shows that (1.1.8) is equivalent to the impedance boundary condition [T18]:

¤ V L τ

t ¡ i

¤ π ψ τ

¥ ¦ x L t e

¡ i

ψ 4 ! ¦ ¥ τ

(1.1.10) L t ¢ e d 2π 0 τ Likewise, (1.1.9) is equivalent to

t ¤

¤ π ψ τ ¦ i x 0 ¥

4 !

¦ ψ ¥ τ

(1.1.11) 0 t ¢ e d 2π 0 t ¡ τ REMARK 1.2 (Inhomogeneous TBC). The (homogeneous) TBC (1.1.9) was derived for mod-

eling the situation where an initial wave function is supported in the computational domain

0 L , and it is leaving this domain without being reﬂected back. If an incoming wave function ¤

ψin t ¦ is given at the left boundary (e.g. a right traveling plane wave), the inhomogeneous TBC

¤ ¤

t ¡ ¤

¤ π ψ τ ψ τ

¦ ¦ 2 d 0 ¥ in

¡ i

4 !

¦ ¥ ¦ ¥ ψ ¥ ψ τ

(1.1.12) 0 t in 0 t ¢ e d

x π dt 0 t ¡ τ

¤ ¤

¦ ¦ ψ ¥ ψ has to be prescribed at x ¢ 0. This is the TBC (1.1.9) formulated for 0 t in t since the TBC was only derived for outgoing wave functions. The inhomogeneous TBC is described and analyzed in detail in [T5].

REMARK 1.3 (Factorization). It should be noted that the Schrodinger¨ equation can formally be factorized into left and right travelling waves (cf. [T6]):

π π ∂ 2 ∂ VL ∂ 2 ∂ VL

¡ i 4 i 4

ψ ¥

£ £ ¢ (1.1.13) e £ i e i 0 ∂x ∂t ∂x ∂t and in the potential–free case:

∂ 2 π ∂ ∂ 2 π ∂

¡ i 4 i 4

¥ ¢ (1.1.14) e £ e 0 ∂x ∂t ∂x ∂t where the term t ¤

d 1 d ψ τ ¦ ! ψ ! τ

(1.1.15) : ¢ d dt π dt 0 t ¡ τ 8 1. THE SCHRODINGER¨ EQUATION 1 can be interpreted as a fractional ( 2) time derivative. 1.2. Well–posedness of the IBVP. We now turn to the discussion of the well–posedness of (1.1.2). The existence of a solution to the 1D Schrodinger¨ equation with the TBCs (1.1.8),

I 1 ¤

¥ ¦ (1.1.9) is clear from the used construction. For regular enough initial data, e.g. ψ ¨ H 0 L ,

ψ ¤ ¦ the whole–space solution x ¥ t will satisfy the TBCs at least in a weak sense. A more detailed discussion is presented in [T9].

It remains to check the uniqueness of the solution, i.e. whether the TBC gives rise to spu-

¥ ¦#" " ψ rious solutions. In order to prove uniform boundedness of t 2 in t we will need L 0 L the following simple lemma which states that the kernel of the Dirichlet–to–Neumann operator π i $ 4 e d % dt is of positive type in the sense of memory equations (see, e.g. [M2]). Using the Plancherel equality for the Laplace transformation (L.4) the following lemma can

be shown: ¤

1 ¤

¨ ¥ ¦ ¦ ©

LEMMA 1.1 ( [T1]). For any T 0, let u H 0 T with the extension u t ¢ 0 for t T. Then

∞ t ¤

π ¤ i d u s ¦

4 ! (1.1.16) Re e u¯ t ¦ ds dt 0 0 dt 0 t ¡ s With this lemma we shall now derive an estimate for the L2–norm of solutions to the

Schrodinger¨ equation (1.1.2). We multiply (1.1.2) by ψ¯ :

i i ¤ 2

¦'& ψ & ¥ ¥ © ψψ ψψ ¥

(1.1.17) ¯ ¢ ¯ V x t 0 x L t 0

t 2 xx Integrating by parts on 0 x L, and taking the real part gives

¤ ¤ ¤

∂ ψ 2 ψ ψ x ) L

& ¥ ¦(& ¥ ¦ ¥ ¦

t t dx ¢ Re i ¯ x t x x t x ) 0 0

¤ V t i L τ

π ¤ V ψ τ ¥ ¦

2 L d L e

¡ i i t

4 !

¦ ψ ¥ τ (1.1.18) ¢ Re e ¯ L t e d π dt 0 t ¡ τ

t ¤

π ¤ ¦ 2 d ψ 0 ¥ τ

¡ i

4 ! τ ¦ Re e ψ¯ 0 ¥ t d π dt 0 t ¡ τ Now integrating in time and applying Lemma 1.1 for the second term and an analogous lemma

for the ﬁrst term yields the estimate

¤ I

" ψ ¥ ¦(" ψ " © *" ¥

(1.1.19) t 2 2 t 0 L 0 L L 0 L

This implies uniqueness of the solution to the Schrodinger¨ IBVP. Equation (1.1.19) reﬂects the ¤

¤ ψ 2

¥ ¦ & ¥ ¦(& fact that some of the initial mass or particle density n x t ¢ x t leaves the computational

ψ ¤ ¥ ¨ " ¦(" domain 0 L during the evolution. In the whole–space problem, x IR, t L2 IR is of course conserved. Finally, we address the question of the well–posedness of the Schrodinger¨ equation (1.1.2) with inhomogeneous TBCs (cf. Remark 1.2). We assume that the incoming wave function ψ ¤ in t ¦ is given at the left boundary by

¤ 2

ω ω

i t x

¦ ¥ ¥ ψ ¥ α ω (1.1.20) in x t ¢ e 0 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 9

i.e. a right traveling plane wave. Then the auxiliary function

¤ ¤

¤ x ¡

ϕ ψ ¡ ψ ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¥ © ¥

(1.1.21) x t : ¢ x t 1 0 t 0 x L t 0 L in fulﬁls the following inhomogeneous Schrodinger¨ equation

¤ ¤

¦ ¥ ¦ ¥ ϕ ∆ϕ ¥ ϕ

£ £ i ¢ V x t f x t

t 2

¤ ¤

¤ x

¡ ¡

¦ ¥ ¦ ¥ ¦ ¥ (1.1.22) ¥ ψ ω

f x t : ¢ 1 0 t V x t

L in

¤ ¤

¤ x ¡

I I ¡

¦ ¦ ¦ ¥

ϕ ¥ ϕ ψ α ¢ x 0 ¢ x x 1 L with the left homogeneous TBC

∂ 2 π ∂ ¤

¡ i 4

¥ ¦ ¥

(1.1.23) e 0 t ¢ 0 ∂x ∂t and the right TBC (1.1.8). Proceeding as in the homogeneous case we obtain

¤ ¤

∂ ϕ ¤ 2 1 ϕ 1 ϕ " ¥ ¦(" " ¥ ¦#" " ¥ ¦("

¯ 2 2

(1.1.24) t Im f dx f t t t 2 L 0 L L 0 L L 0 L

" ¥ ¦(" If we further assume f t 2 F , t 0 then it follows easily that L 0 L

¤ I L F

¥ ¦#" *" " ¥ © ¥ " ψ ϕ α

(1.1.25) t 2 2 £ t t 0 L 0 L L 0 L 3 2 and this implies the well–posedness.

2. Discrete Transparent Boundary Conditions In this Section we shall discuss how to discretize the TBC (1.1.8) in conjunction with a Crank–Nicolson ﬁnite difference scheme and review the derivation of the DTBC from [T1].

∆ ∆ ψn ψ ¤

¥ ¦ ¢ With the uniform grid points x ¢ j x, t n t, and the approximations x t the j n j + j n discretized Schroding¨ er equation (1.1.1) reads:

2 1 1 1 1

2 n n n n

n ¡ 2 2 2 2

¥ ¦ ¥ ψ ψ ψ ¥

£ ¢

(1.2.1) i D ¢ D V V V x j t 1 t j x j j j j n

2 2

n 1 2 n 1 n

ψ ψ ψ ¦ % £ with the time averaging ¢ 2. Here Dt denotes the forward difference j j j 2

quotient in time and Dx is the second order difference quotient in space, i.e.

ψn 1 ¡ ψn ψn ψn ψn

2 £

n j j 2 n j 1 j j 1

ψ ¥ ψ ¢ D ¢ D t j x j 2 ∆t ∆x ¦

REMARK 2.1. Most existing discretization schemes for the Schrodinger¨ equation with TBCs are also based on the Crank–Nicolson ﬁnite differences ( [T6], [T16], [T19]). For our analysis, one of the main advantages of this second order (in ∆x and ∆t) scheme is, that it is unconditionally stable [F8] and an easy calculation shows that it preserves the discrete

2 n 2 n 2

ψ " ∆ & ψ &

" ∑ ¢ L –norm: 2 x j , ZZ j , which is the discrete analogue of the mass conservation property of (1.1.1).

10 1. THE SCHRODINGER¨ EQUATION $

n 1 2

¡ ψ % In order to derive this discrete mass conservation property we multiply (1.2.1) with i ¯ j :

1 1 1 1 1 2

n i n 2 n i n n ψ 2 ψn ψ 2 ψ 2 ¡ 2 ψ 2

(1.2.2) ¯ D ¢ ¯ D V j t j 2 j x j j j

Summing it up for j ¨ ZZ (i.e. in absence of boundary conditions) gives with summation by parts

1 1 2 1 1 2

n i n i n n ¡ ψ 2 ψn ¡ ψ 2 2 ψ 2

(1.2.3) ∑ ¯ D ¢ ∑ D ∑ V

j t j 2 x j j j

, , j , ZZ j ZZ j ZZ

Finally, taking the real part by using the simple identity (“discrete product rule”)

¤ ¤

¤ 1 1

ψn υn ψn 2 υn υn 2 ψn

¦ ¦ ¦ ¥ £

(1.2.4) D ¢ D D

t t t υn ψn i.e. with ¢ ¯

1 .

ψn 2 ψn 2 ψn & & - ¥

(1.2.5) D ¢ 2Re ¯ D t t yields the conservation of the mass :

ψn 2

(1.2.6) D ∑ ¢ 0

t j

j , ZZ

REMARK 2.2. We remark that an arbitrary high (even) order conservative scheme for the Schrodinger¨ equation (1.1.1) can be obtained by using the diagonal Pade´ approximations to the exponential [F1]. The Crank–Nicolson scheme corresponds to second order, and the fourth order is known in the ODE literature as Hammer and Hollingsworth method [F2]. 2.1. Discretization strategies for the TBC. We shall now compare four strategies to discretize the TBC (1.1.8) with its mildly singular convolution kernel. First we review a known discretization from the literature, where the analytic TBC in the equivalent form (1.1.10) at ∆

L ¢ J x was discretized in an ad–hoc fashion. Discretized TBC of Mayﬁeld. In [T16] Mayﬁeld proposed the approximation

¤ V L τ t ¡ i n 1 t

n ¤ V m 1

¦ ψ ¥ τ τ

x L tn e 1 L ∆ d

n m ¡ n m i m t

! !

τ ψ ψ ¦ / ∑ τ d ∆ J J 1 e τ 0 x tm m ) 0

(1.2.7) ! ¤ V L ∆

n 1 n m ¡ n m i m t

ψ ψ ¦

∆

2 t J J 1 e

¥ !

¢ ∑

∆x £ m £ 1 m m ) 0 where she used the left–point rectangular quadrature rule. This leads to the following discretized TBC for the Schrodinger¨ equation: n 1

∆ 0

¡ ¡ n ¡ n n n m n m

ψ ψ ! ψ ψ ψ ˜ ¥

(1.2.8) J J 1 ¢ J ∑ J J 1 m 2B ∆t m ) 1 with V

i L m∆t

π 0 e

¡ i

4 ˜ !

! ¢ B ¢ e

2π m £ m £ 1 m On the fully discrete level this BC is no longer perfectly transparent. For the resulting scheme

with a homogeneous Dirichlet BC at j ¢ 0 and (1.2.8), Mayﬁeld obtained the following result: 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 11

THEOREM 1.2 ( [T16]). The numerical scheme (1.2.1), (1.2.8) is stable, if and only if

¤ ∆ ¤

t 2 2

¨ ¦ ¥ ¦

(1.2.9) 4 2 j £ 1 2 j ∆x2 j , IN0 This shows that the chosen discretization of the TBC (1.2.8) destroys the unconditional stability of the underlying Crank–Nicolson scheme. The stability regions of Theorem 1.2 are illustrated in Figure 1.2 as light areas. The dark intervals are regions of instability.

∆__t C ∆x2

__1 __1 __1 1 16 9 4

FIGURE 1.2. Discretized TBC of Mayﬁeld : Stability regions.

As the numerical example in Section 2.5 will show this discretization gives rise to unphysi- cal reﬂections at the boundary.

REMARK 2.3 (Approach of Baskakov and Popov). A similar strategy using a higher-order quadrature rule for the l.h.s. of (1.2.7) was introduced by Baskakov and Popov in [T6]. This approach typically induces less numerical reflections compared to the results when using the discretized TBC of Mayfield. Semi–Discrete TBC of Schmidt and Deuflhard. In the semi–discrete approach of Schmidt and Deuflhard [T23] a TBC is derived for the semi–discretized (in t) Schrodinger¨ equation. This

method also applies for a nonuniform in t (e.g. adaptive) interior scheme and it admits a time–

¤ ¦ dependent potential in the exterior domain (i.e. VL ¢ VL t ). While being unconditionally stable (in conjunction with an interior finite element scheme) [T24], it still exhibits small residual reflections at the artificial boundary. In [T24] this approach is also applied to uniform exterior z–discretizations, and one then recovers — through a different derivation — the discrete TBC from [T1].

Approach of Lubich and Schadle.¨ The time discretization is done by the trapezoidal rule in the interior and by convolution quadrature on the boundary. The numerical integration of the ˆ ¤ convolution integral is done in the following way (cf. [T14], [T15] [T22]). If f s ¦ denotes the ∂

Laplace transform of f , then formally setting s ¢ t yields

∞

¤ ¤ ¤ ¤ τ∂

ˆ t ¦ ¦ ¦ ∂ ¦ τ τ

f t g t ¢ f e g t d 0

∞

¤ ¤

(1.2.10) τ ¡ τ τ

¦ ¦

¢ f g t d

1 ¥

¢ f g

¤ ¤

¤ ˆ ∂

¦ ¦ ¦ where g is a function satisfying g t ¢ 0 for t 0. Now f t g t is approximated by the

discrete convolution

¤ ¤ ¤

ˆ ∂k ω ¡

¦ ¦ ¦ ¥ (1.2.11) f t g t : ¢ ∑ n g t nk

n 2 0 12 1. THE SCHRODINGER¨ EQUATION ∆ ω with the step size k ¢ t. The quadrature weights n are deﬁned as the coefﬁcients of the

generating power series:

δ ξ ¦

n ˆ & & ω ξ ¥ ξ

(1.2.12) ∑ : ¢ f small n k

n 2 0

¤ ¤

δ ξ ¦ δ ξ ¦

Here is the quotient of the generating polynomials of a linear multistep method, e.g. ¢

¤ ¤ ¤

¡ ¡

¦ % ¦ ξ δ ξ ¦ ξ ξ ¢ 1 for the implicit Euler method and 2 1 1 £ for the trapezoidal rule. If one chooses for the quadrature the same numerical scheme as in the interior then one obtains also a reﬂection–free discrete TBC. Discrete TBC. Instead of using an ad–hoc discretization of the analytic TBC like (1.2.7) we will construct discrete TBCs of the fully discretized whole–space problem. Our new strat-

egy solves both problems of the discretized TBC at no additional computational costs. With our DTBC the numerical solution on the computational domain 0 j J exactly equals the discrete

whole–space solution (on j ¨ ZZ) restricted to the computational domain. Therefore, our overall scheme prevents any numerical reﬂections at the boundary and inherits the unconditional stability of the whole–space Crank–Nicolson scheme (see Theorem 1.7). These different approaches, discretization of the analytic TBC and (semi–)discrete TBC, are sketched in Figure 1.3. .. Schrodinger Equation

.. (Analytic) Transparent BC Discrete Schrodinger Equation

Discretized TBC semi-discrete TBC Discrete TBC Mayfield Schmidt & Deuflhard Arnold & Ehrhardt .. Baskakov & Popov Lubich & Schadle only conditionally stable unconditionally stable unconditionally stable numerical reflections small reflections reflection-free same computational effort

FIGURE 1.3. Discretization strategies for the TBC

Consequently, when considering the discretization of TBCs, it should be a standard strategy to derive the discrete TBCs of the fully discretized problem, rather then attempting to discretize the differential TBC whenever it is possible. A comparison of these two strategies for a 1D wave propagation problem is given in [T10]. 2.2. Derivation of the DTBC. To derive the discrete TBC we will now mimick the derivation of the analytic TBC from Section 1 on a discrete level. The Crank–Nicolson scheme (1.2.1) can be written in the form:

¤ ¤

2 2 ¡

¡ n 1 n n 1 n 2 n 1 n

¦ ¥ ψ ψ ¦ ∆ ψ ∆ ψ ψ ψ

¢ £ £ (1.2.13) iR j j x j £ x j wVj j j 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 13

with

¤ ¤

2 2 1

¦ ∆ ¦ ∆ 4 x 2 x n

¡ 2

¥ ¥ ¥ ¦ ¥

¢ ¢ R ¢ w V V x j t 1 j n ∆t 2 2

∆2 ψn ψn ¡ ψn ψn £ where ¢ 2 , and R is proportional to the parabolic mesh ratio. x j j 1 j j 1

Again, we will only consider the right BC. In analogy to the continuous problem we assume ¡

n ¡ 0

¢ ¢ for the potential and initial data: Vj ¢ VL const, j J 1, j 0, j J 2 and solve the discrete right exterior problem by using the discrete analogue of the Laplace transformation, the –transform :

∞

n . ψn n ψ ¦ ¥ ¨ ¥ & &3© ¥ - ψ ¢ (1.2.14) j ¢ ˆ j z : ∑ j z z CI z 1

n ) 0

which is described in more detail in the Appendix. Hence, the –transformed Crank–Nicolson ¡

ﬁnite difference scheme (1.2.13) for j J 1 reads

¤ ¤ ¤ ¤ ∆

2 t VL ¡

¡ ψ ¦ ∆ ψ ¦ κ ¦ ¦ ¥ κ

¢ £ £ ¢ (1.2.15) z £ 1 ˆ z iR z 1 i z 1 ˆ z x j j 2

The two linearly independent solutions of the resulting second order difference equation

¤ ¤

¤ iR z 1

¡ ¡ ¡

¦ ¦ ¥ ¥ ψ ¦ κ ψ ψ

£ £ ¢

(1.2.16) ˆ j 1 z 2 1 i ˆ j z ˆ j 1 z 0 j J 1

2 z £ 1

¤ ¤ ¤ j

ψ ν ¡ ν

¦ ¦ ¦ ˆ ¢

take the form j z 1 2 z , j J 1, where 1 2 z solve ¡

iR z 1 ¡

ν2 ¡ κ ν

£ ¢ (1.2.17) 2 1 £ i 1 0

2 z £ 1

¤ ¤

∞ ν ¤ ν ν & ¦(&4 ¦ ¦

For the decreasing mode (as j ) we have to require 1 z 1 and obtain (using 1 z 2 z ¢

1) the –transformed right DTBC as

¤ ¤ ¤

¦ ¦ ψ ¦ ν ψ

(1.2.18) ˆ J 1 z ¢ 2 z ˆ J z

Analogously, the –transformed left DTBC reads:

¤ ¤ ¤

¦ ¦ ¥ ψ ¦ ν ψ

(1.2.19) ˆ 1 z ¢ ˜ 2 z ˆ 0 z ¤

¤ ν & ¦#&5© where ν˜ 2 z ¦ with ˜ 2 z 1 is obtained from a solution to the left discrete problem, i.e. (1.2.16)

on the range j 1. ψ ¤

REMARK 2.4 (Discrete Factorization). If S denotes the usual shift operator given by S ˆ j z ¦

¤ ψ ¦

¢ ˆ z , then analogously to the continuous case (cf. Remark 1.3) the discretized Schrodinger¨ j 1

equation (1.2.16) can formally be factorized as:

¤ ¤ ¤

¡ ¡ ¡

ν ¦ ¦ ¥ ¥ ν ¦ ψ

(1.2.20) S 1 z S 2 z ˆ j 1 z ¢ 0 j J 1 which leads to the same DTBCs (1.2.18), (1.2.19). It remains to inverse transform (1.2.18) using the inversion rules of the –transform given

in the Appendix. By the following tedious calculation this can be achieved explicitly.

¤ ¤

1 .

¦ ¦

- ν ν

CALCULATION (of 2 z ). First we rewrite 1 2 z as:

¡ ¡ ¡

¤ iR z 1 iR z 1 iR z 1

¡ ¡ ¡

ν ¦ κ κ κ

¢ £ £ £

1 2 z 1 i i 2 i

£ £ (1.2.21) 2 z £ 1 2 z 1 2 z 1

iR Rκ z iR 1 ¡

¡ 2

£ £ £

¢ 1 iR Az 2Bz C

7 £ 2 2 z £ 1 2 z 1 14 1. THE SCHRODINGER¨ EQUATION with the constants

¤ κ κ 4 ¦ ¥

£ £ £ (1.2.22a) A ¢ 1 i 1 i i R κ

κ2 4 ¥

£ £ (1.2.22b) B ¢ 1

¤ 4

¡ ¡

¡ κ κ ¦

(1.2.22c) C ¢ 1 i 1 i i R For the inverse –transform we use

2 ¡ z

1 Az 2Bz £ C C

2 ¡

8 ¢

(1.2.23) Az 2Bz £ C

A z A 2 ¡ B

z 2 z £ 1

C C

With the abbreviations !

¤ z A B

! !

¥ ¦ ¥ λ ¥ ¥

¢ ¢ (1.2.24) F z µ ¢ µ

2 ¡

z 2µz £ 1 C A C

we obtain from equation (1.2.23)

2 ¡ ¤

1 1 Az 2Bz £ C

2 ¡

! ¤

λ ¥ ¢

Az 2Bz £ C F z µ

¦ £

z £ 1 A z z 1

1 C E ¤

λ ¥ £ (1.2.25) ¢ A F z µ

A z z £ 1

! 1 E 1 ¤

1 ¡

λ λ λ

¥ ¦ ¥ £

¢ C F z µ

z A C z £ 1

£ £

with E ¢ A 2B C. The inversion rules now yield

! ¡

2 !

¤ ¤

Az 2Bz £ C E

1 0 0

¡ ¡

1 1 ¡ n !

λδ λ δ ! δ ¦ 1 ¦ £ ¢ C 1 P µ n n n n

z £ 1 A C

! n 1

¤ ¤ ¤ ¤

E ¡

1 ¡ n k

λ ¦ λ ¦ ¦ ¦ ¥ £ ¢ C P µ P µ ∑ 1 P µ n n 1 k

A C

) k 0

where 1 denotes the discrete convolution. Finally, we obtain ¤ ¤ κ R R

1 . 0 ¡

¡ n

ν δ

- ¦ ¦

¢ £ £ z 1 i iR 1 1 2 2 2 n

(1.2.26) ! n 1

¤ ¤ ¤ ¤

iR C E

n ¡ n k

! !

λ λ λ

¦ ¦ ¦ ¦

Pn µ £ Pn 1 µ ∑ Pk µ

7 2 A C

) k 0

¯ &

& λ ¢

Since C ¢ A we have 1 with

¤ ¡

¡ 2

κ κ κ ¦ £

A R 4 R £ 2i R 2 !

λ ! ¢

(1.2.27) ¢ ¤

A C ¤ 2 2 2

¦ κ ¦ κ

£ £ 1 £ R R 4

Therefore we write

κ ¦

λ iϕ ϕ 2 R £ 2 ¥ ¢

(1.2.28) ¢ e with arctan ¡ R ¡ 4κ Rκ2 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 15 We obtain for the parameter µ:

¤ 2

κ ¦ κ £

R 1 £ 4

¨ ¥

(1.2.29) µ ¢ IR ¤

¤ κ2 2 κ 2 ¦ ¦

£ £

1 £ R R 4

¡ and it can easily be seen that 1 µ 1 is valid. The constant E simply equals 4 and we get

E 4R

τ ¨ ¢

(1.2.30) ¢ IR ¤

A C ¤ 2 2 2

κ κ ¦ ¦

£ £ 1 £ R R 4

The choice of the sign in (1.2.26) can be justiﬁed analytically or simply by testing it numer-

n 1 .

- ν

ically. We ﬁnally obtain the convolution coefﬁcients ¢ 2 z as

¤ ¤ ¤

0 σ

R i ϕ 9

0 4 $

¡ ¡

n ¡ n 2 2 2 2 i 2

δ ¦ σ ¦ σ ¦

£ £ £ £ £ ¢ 1 i iR 1 R R 4 e 2 2 n 2

(1.2.31) n 1

¤ ¤ ¤ ¤

ϕ 9 ¡

in ¡ n k

λ ¦ ¦ τ λ ¦ ¦ ¥

e Pn µ £ Pn 1 µ ∑ Pk µ

k ) 0 σ κ

with ¢ R and Pn denotes the Legendre polynomials. The resulting discrete TBCs at the grid

¥ ¥ points x j j ¢ 0 J read

n 1

0 0

0 n k

n ¡ n k

¥ ψ ψ ψ ¥ (1.2.32a) 1 0 0 ¢ ∑ 0 0 n 1

k ) 1

n 1

0 0

0 n k

n ¡ n k

ψ ψ ψ ¥

(1.2.32b) J 1 J J ¢ ∑ J J n 1

k ) 1

0 The subscript j of the coefﬁcients n indicates at which boundary the values are to be taken. In the sequel many parameters will be supplied with this subscript.

2.3. The Asymptotic Behaviour of the Convolution Coefﬁcients. We study the asymp-

0 0

n n totic behaviour of the 0 , J in (1.2.32). It will turn out that it is advantageous to reformulate the DTBC using new coefﬁcients. Afterwards we shall derive a recursion formula for these new coefﬁcients and compare their decay rate with the decay rate of the continuous integral kernel in (1.1.8), (1.1.9).

The summed convolution coeffcients. First we want to study the asymptotic behaviour of

¥ θ θ π ¢ the convolution coefﬁcients j , j ¢ 0 J. With the notation µ j cos j, 0 j , we use the following classical result on the asymptotic property of the Legendre polynomials:

LEMMA 1.3 (Theorem 8.21.2 (Formula of Laplace), [S7]). ! ¤ π

1 ¡

¤ ¤ ¦ θ 2 cos n £ j

2 4 3 $ 2

¦ ¥ θ ¦ θ π

£ (1.2.33) Pn cos j ¢ O n 0 j π sinθ j n

ε θ π ¡ ε

The bound for the error term holds uniformly in the interval j . ¤

∞ ¦ : From this lemma we conclude that limn Pn µ j ¢ 0 holds. Consequently, the coefﬁcients ∞ have the following asymptotic behaviour for n :

n 1

¤ ¤ ¤ ¤ ¤ ¤

0 τ

i ϕ ϕ

n j $

¡ ¡ ¡

¡ n 4 2 2 2 2 i j 2 n i j k

¦ σ ¦ σ ¦ ¦ ¦ ¦

£ £ (1.2.34) iR 1 R £ R 4 e 1 ∑ e P µ j + 2 j j k j

k ) 0 16 1. THE SCHRODINGER¨ EQUATION Using

n 1

¤ ¤ ϕ 1 1 1

¡ i j k

¦ ¦ ¢ lim ∑ e P µ ¢ ∞ k j n : 2 λ

Re £ µ

k ) 0 λ¯ λ¯ 2λ¯ j j £

(1.2.35) 1 £ 2µ j j j j ¤ 1 ¤

4 2 2 2 2 iϕ j $ 2

¦ ¥

σ ¦ σ

£ £ £ ¢ R R 4 e 2R j j

we ﬁnally obtain

¤ ¤ ¤ ¤ ¤

0 τ

n i j

¡ ¡

¡ n 2 σ2 2 σ 2 n n

¦ ¦ ¦ ¦ ¦

£ £ £ ¢ (1.2.36) iR 1 £ R R 4 1 2iR 1

j + 4R j j

0 n The sequence j is asymptotically an alternating, purely imaginary sequence, which may lead (on the numerical level) to subtractive cancellation in (1.2.32). To circumvent this problem we

consider the summed coefﬁcients

0 0 0

n n n 1 0 0

¥ ¥ ¥ ¥ ¥

£ ¢ ¢ (1.2.37) s j : ¢ j j n 1 s j : j j 0 J

and compute:

¤ ¤

0 σ R i

0 j ϕ $ ¡

¡ 4 2 σ2 2 σ 2 i j 2

¦ ¦

£ £ £ £ j ¢ 1 i R j R j 4 e (1.2.38) 2 2 2

R σ j i ¤

¡ ¡ ¡

¡ 2 2

σ σ σ ¦ ¥

£ £ £ ¢ 1 i R 4 j 2iR j 2

2 2 2 j

¤ ¤

0 i

1 ϕ $ ϕ

¡ 4 2 2 2 2 i j 2 i j

¦ ¥

σ ¦ σ τ

£ £ £ £ £ (1.2.39) ¢ iR R R 4 e µ e j 2 j j j j

and for n 2 we compute using the deﬁnition of E:

¤ ¤ i ϕ ϕ n $

¡ 4 2 2 2 2 i j 2 in j

¦ σ ¦ σ

£ £ £ s ¢ R R 4 e e

j 2 j j

¤ ¤ ¤

(1.2.40) 1 ¡

¦ ¦ ¦ λ λ τ

£ £

Pn µ j £ j j j Pn 1 µ j Pn 2 µ j

) 2µ j

With the recurrence relation of the Legendre polynomials

¤ ¤

¤ n n 1

¦ ¦ ¦ ¥ ¥ £

µ jPn 1 µ j ¢ Pn µ j Pn 2 µ j n 1

¡ ¡ 2n 1 2n 1

we ﬁnally get

¤ ¤

¦ ¦ i ϕ ϕ P µ P µ n 4 2 i j $ 2 in j n j n 2 j

2 2 2

σ ¦ σ ¦

(1.2.41) ¥

£ £ £ s ¢ R R j 4 e e n 2

j 2 j 2n ¡ 1

0 0

REMARK 2.5. The coefﬁcient j can also be calculated with [S3, Theorem 39.1] by

¤ ¤ ¤ ¤

0 σ

0 R j i

¡ ¡

ν ¦ σ ¦ σ ¦§¦

¢ £ £ £ £ (1.2.42) ¢ lim 2 z 1 i R i j R i j 4 j ∞ z : 2 2 2 We summarize our results in the following theorem:

2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 17 ¢ THEOREM 1.4 ( [T1]). The left (at j ¢ 0) and right (at j J) discrete TBCs for the Crank– Nicolson discretization (1.2.1) of the 1D Schroding¨ er equation are respectively

n 1

0 n k ¡

n ¡ n k n 1

ψ ψ ψ ψ

¥ ¥ (1.2.43a) 1 s0 0 ¢ ∑ s0 0 1 n 1

k ) 1

n 1

0 n k ¡

n ¡ n k n 1

¥ ¥ ψ ψ ψ ψ

(1.2.43b) J 1 sJ J ¢ ∑ sJ J J 1 n 1

k ) 1

with

¤ ¤

σ σ ¦ ¦ n R j 0 R j 1 ϕ Pn µ j Pn 2 µ j

¡ in j

δ δ α ¥

£ £ £ £ £ (1.2.44) s ¢ 1 i n 1 i n j e j 2 2 2 2 2n ¡ 1

¤ 2 σ σ2

σ ¦ £ R £ 4

2R j £ 2 j j

ϕ ¥ ¥ ¢

j ¢ arctan µ j

¤ ¤ ¡ 2 ¡ 2

R 4σ j σ 2 2 2 2 ¦

j σ ¦ σ

£ £ R £ j R j 4

¤ 2∆x i ¤ ϕ

4 2 2 2 2 i j $ 2

σ ¥ α σ ¦ σ ¦ ¥ ¥

¢ £ £ £ ¢ ¢ V R R 4 e j 0 J j 2 j j 2 j j

δ j ;

Pn denotes the Legendre polynomials (P 1 ; P 2 0) and n the Kronecker symbol.

¨ ¥ ¦ The Pn only have to be evaluated at the two values µ0 ¥ µJ 1 1 , and hence the numerically stable recursion formula for the Legendre polynomials can be used [E2].

The recurrence formula for the summed coefﬁcients. In this subsection we shall give n

two different derivations for the recursion formula of the convolution coefﬁcients s j . The n

ﬁrst one is based on the explicit representation (1.2.41) of s j by ﬁrst calculating a recursion

¤ ¤

¦ ¦

formula for Pn 1 µ j Pn 1 µ j . The second derivation does not require the explicit form of the

n ν ¦ coefﬁcients s j but only the growth functions 1 2 z from the –transformed DTBCs (1.2.18) and (1.2.19).

FIRST DERIVATION: Here we start with the standard recursion formula [S1] for the Le-

¤ ¤

¦ ¦

gendre polynomials Pn 1 µ j , Pn 1 µ j :

¤ ¤ ¤ ¤ ¤

¦ ¦ ¦ ¦ ¦ ¥ ¥

£ £

n 1 Pn 1 µ j ¢ 2n 1 µ jPn µ j nPn 1 µ j n 0

¤ ¤ ¤ ¤ ¤

(1.2.45) ¤

¡ ¡ ¡ ¡

¦ ¦ ¦ ¦ ¦ ¦ ¥ ¥

n 1 Pn 1 µ j ¢ 2n 3 µ jPn 2 µ j n 2 Pn 3 µ j n 2

and

¤ ¤

¤ ¤ ¤ ¦

Pn 3 µ j ¦ Pn 1 µ j

¦ ¦ ¦ ¥ ¢

(1.2.46) Pn 1 µ j 2µ jPn 2 µ j £ Pn 3 µ j n 2

2n 3

¤ ¤

¦ ¦

The expression Pn 1 µ j Pn 1 µ j is converted in the following way:

¤ ¤ ¤

¦ ¦ ¦

n £ 1 Pn 1 µ j Pn 1 µ j

¡ ¡

¤ ¤ ¤ ¤ ¤

¤ 2n 3 n 2

¦ ¦ ¦ ¦ ¦ ¦

¢ £ £ £

2n 1 µ jPn µ j 2n 1 µ jPn 2 µ j £ 2n 1 Pn 3 µ j

¡ ¡

n 1 n 1

¤ ¤

¤ ¤ ¤ ¤ ¤

¦ ¦

n 2 2n £ 1

¡ ¡ ¡

¦ ¦ ¦ ¦ ¦ ¥ £

¢ 2n 1 µ j Pn µ j Pn 2 µ j Pn 1 µ j Pn 3 µ j

2n 3 18 1. THE SCHRODINGER¨ EQUATION

where we have used the relation

¤ ¤

¤ ¤ ¤ ¤ ¦

1 Pn 3 µ j ¦ Pn 1 µ j

¡ ¡

¦ ¦ ¦ ¦ £

Pn 3 µ j µ jPn 2 µ j ¢ Pn 1 µ j Pn 3 µ j

2 2n 3

n 1 ¤

¡ ¡

¦ ¦

¢ Pn 1 µ j Pn 3 µ j

2n 3

¤ ¤

¦ ¦

Finally, we obtain for n 2 the following recursion formula for Pn 1 µ j Pn 1 µ j :

¤ ¤

¦ ¦ ¢

(1.2.47) Pn 1 µ j Pn 1 µ j

¤ ¤

¤ ¤ ¤ ¤

¦ ¦ £

2n £ 1 n 2 2n 1

¡ ¡ ¡

¤ ¤

¦ ¦ ¦

µ j Pn µ j ¦ Pn 2 µ j Pn 1 µ j Pn 3 µ j

¦ ¦ £

n £ 1 n 1 2n 3 n Since the s j are determined by

¤ ¤

n j

α ¦ ¦ ¥ ¥

(1.2.48) s ¢ j Pn µ j Pn 2 µ j n 2

j ¡ 2n 1

we get from (1.2.47) the recurrence relation for the summed convolution coefﬁcients:

¡ ¡

n 1 2n 1 n n 2 n 1

1 ¡ 2

λ λ ¥ ¥

(1.2.49) s j ¢ µ j j s j j s j n 2 £

n £ 1 n 1

¥ ¥ which can be used after calculating the ﬁrst values s j for n ¢ 0 1 2 by the formula (1.2.44).

SECOND DERIVATION: Next we shall present an alternative derivation of the ﬁrst convolu-

¥ ¥

tion coefﬁcients s j , n ¢ 0 1 2 and the recurrence relation (1.2.49). The advantage of this alter- ¤

ν ¦ native approach is that we shall only need the growth functions 1 2 z from the –transformed DTBCs (1.2.18) and (1.2.19). Hence this approach might also apply to a bigger class of linear evolution equations, where it is not possible (or too tedious) to derive an explicit representation of the convolution coefﬁcients.

We remark that an even more advantageous approach might be based on the polynomial ¤

equation (1.2.17) for the growth function ν z ¦ , rather than on its explicit solution. The beneﬁt of such a strategy would lie in the possibility to obtain the convolution coefﬁcients also for higher order difference schemes, that would lead to quartic (or even higher order) equations for ν ¤

z ¦ . To our knowledge this has, however, not been accomplished yet. ¤

In this second approach we shall ﬁrst derive a ﬁrst order ODE for the growth function ν z ¦ , which is explicitly given by (1.2.21). In fact, it is more convenient to consider

∞

¤ ¤ ¤

n 1 n

ν ν

¦ ¦ ¦

£ ¢ (1.2.50) ˜ z : ¢ z 1 z ∑ s j z

n ) 1

Using the constants (1.2.22) and (1.2.24) it has the explicit form

¤ iR σ j iR σ j i λ 1 ¡

ν ¡ 2 σ σ2 2 ¦ 6

£ £ £ £ £ £ £

(1.2.51) ˜ 1 2 z ¢ z 1 1 R 4 j z 2z

2 2 2 2 2 j µ λµ ¥

The index j ¢ 0 J again denotes the grid point where the DTBC is to be constructed. Multiply-

dν˜ λ 1 ¤

ν 2 ¡ ν < ¦ £ ing ˜ ¢ by z 2z then yields an inhomogeneous ﬁrst order ODE for ˜ z :

dz µ λµ

¤ ¤

λ ¤ λ

¡ ¡

2 ¡ 1 0

¦ ¦ ¥

ν ¦ ν β β β

¢ ¢ £ (1.2.52) z 2z £ ˜ z z 1 ˜ z z : z µ λµ µ 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 19 with

λ σ σ

iR j iR j

¡ ¡

1 ¡

£ £ £ £ ¢ 1 1 µ 2 2 2 2 (1.2.53)

σ σ 1 iR j iR j

β 0 ¡

£ £ £ £ ¢ 1 1

λµ 2 2 2 2

¤ ν ¦

Its general solution includes ˜ 1 2 z as deﬁned in (1.2.51). Using the Laurent series (1.2.50) of ν ν n

˜ and ˜ < in (1.2.52) immediately yields the desired recursion for the coefﬁcients s j :

1 0 ¡

¡ β 1 ¥

s s ¢

µ j j

λ 2 1 1 0 ¡ β 0

(1.2.54) ¥

£ ¢ 2 s £ s s

µ j j λµ j

¤ ¤ ¤

n 1 µ n 1 n 1

¡ ¡ ¡ ¡

¦ ¦ ¦ ¥ ¥

£ ¢ n £ 1 s 2n 1 s n 2 s 0 n 2 j λ j λ2 j which coincides with (1.2.49).

The starting coefﬁcient of the recursion can be determined as in (1.2.42): ¤

λ 0 ν˜ z ¦ R j i

¡ 2 2

6 σ σ

¢ £ £ £ s ¢ lim 1 i R 4 j j j

z : ∞ z 2 2 2 µ

¤ ¤

ν ¦(&= & ν ¦(&=© Here, the sign has to be ﬁxed such that & 1 z 1 for the right DTBC and 2 z 1 for the ∞ left DTBC. This can be done for e.g. for z ¢ .

STABILITY OF THE RECURRENCE RELATION: For proving that the recurrence relation (1.2.49) is well–conditioned we follow the notation in [E2] and write (1.2.49) as the second

order difference equation

n 1 n n n n 1

¥ ¥

£ ¢ (1.2.55) s j £ a j s j b j s j 0 n 2

with

¡ ¡

n 2n 1 n n 2

¡ 1 2

λ λ

¢ ¢ (1.2.56) a ¢ µ b 0

j j j j j > £

n £ 1 n 1

There are two linearly independent solutions s n , s n to (1.2.55). If they have the property

j 1 j 2 s n j 2

(1.2.57) lim ¢ 0 ∞ n : s n

j 1 then s n is called a minimal solution and serious numerical problems arise if one tries to com-

j 2 pute the solution s n in a straightforward way by using the recursion (1.2.49). If there is an j 2

error in the initial data but recurring with inﬁnite precision then the relative error of the intended approximation to s n becomes arbitrarily large. Methods of calculating minimal solutions of j 2 three–term recurrence relations can be found in [E2]. To prove that (1.2.49) is well–conditioned we have to show that the seeked solution is not a minimal solution to (1.2.55). This type of solution is called dominant.

20 1. THE SCHRODINGER¨ EQUATION

n n

Since the coefﬁcients a j , b j in (1.2.55) have the ﬁnite limits

n B j n Cj ¡

¡ 1 2

¥ ¥ ¥ ¥ λ λ

¢ ¢ ¢ ¢ ¢ ¢

(1.2.58) a j ¢ lim a j 2µ j j 2 b j lim b j j j 0 J

: : ∞ ∞ n A j n A j one calls (1.2.55) a Poincare´ difference equation and

Φ ¤ 2 ¦

£ £ (1.2.59) j t ¢ t a jt b j

the characteristic polynomial of (1.2.55). The characteristic polynomial has the complex conju-

1 2 1 2

6 % ¦ % & & ¢ gate zeros t j ¢ B j i4 R A j. The zeros have the same moduli: t j 1, and therefore the classical Theorem of Poincare´ (formulated below for the special case of a second–order difference equation) cannot be applied to distinguish two solutions with distinct asymptotic

properties.

1 2

THEOREM 1.5 (Poincare´ Theorem, [E1]). Suppose that the zeros t j , t j of the character- n

istic polynomial (1.2.59) have distinct moduli. Then for any nontrivial solution s j of (1.2.55)

n 1 s j k

lim ¢ t ∞ j n : n

s j ¢ for k ¢ 1 or k 2.

REMARK 2.6. If equation (1.2.55) has characteristic roots with equal moduli then Poincare’´ s

Theorem may fail (cf. example of Perron [E1]). ¤

It is well–known that the Legendre polynomials Pn µ j ¦ and the Legendre functions of the ¤ second kind (of order zero) Qn µ j ¦ satisfy the same three–term recurrence relation (1.2.45).

Therefore, the two linearly independent solutions to (1.2.55) are the convolution coefﬁcients

n n n

s ¢ s (1.2.48) and s given by j 1 j j 2

λ n

¤ ¤ n j

β ¡

¦ ¦ ¥ ¥

(1.2.60) s ¢ Q µ Q µ n 2 j n j n 2 j

j 2 ¡ 2n 1 with some constant β j. Now we want to study the asymptotic behaviour of these two solutions. With the notation

θ θ π

µ j ¢ cos j, 0 j , we use Lemma 1.3 which gives !

¤ π ¡

θ ¡ 1

¤ ¤ ¤ θ

2 2 sin j sin n ¦ j

$ ¡

¡ 2 4 3 2

! !

θ θ

¦ ¦ ¦ ¥ £ (1.2.61) Pn cos j Pn 2 cos j ¢ O n π n

and from (1.2.48) we see that !

¤ π ¡ ¡ 1

θ θ

¦ n 2 sin j sin n j

¡ n 2 4

α λ ¥ ∞ ! (1.2.62) s ¤ n j + j j π ¡ 1

n 2 ¦ n ¤

An analogous formula to (1.2.33) for the Legendre functions of the second kind Qn µ j ¦ is given by the following lemma:

θ π

LEMMA 1.6 (Theorem 8.21.14, [S7]). For 0 j ¤

! 1 π

¤ ¤ θ

π ¦ £ cos n £ j

2 4 3 $ 2 !

θ ! ¦ ¦ £ (1.2.63) Qn cos j ¢ O n 2 sinθ j n

2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 21

¥ This holds uniformly in the interval ε π ε .

As before we can deduce from (1.2.60) that ! ¤ π

π θ ¡ 1 θ

n sin j sin n j £

¡ n 2 4

¥ β λ ∞ !

(1.2.64) s ¤ n + j 2 j j

¡ 1 2 n 2 ¦ n

holds. Therefore the ratio of the two solutions behaves asymptotically as

n ¤ π

¡ 1

θ ¤

s β π ¦ β π π £ j 2 j sin n j j 1

2 4 ¡

¥ ¥

¦ θ ∞

¤ £

¢ tan n n

π j ¡ n α ¡ 1 θ α j 2 sin n ¦ j 2 2 4

s j 2 j 4

i.e. neither s n nor s n can be a minimal solution. Consequently, the problem of determining the j j 2 required values of the convolution coefﬁcients is well–conditioned [E2]: they can be computed numerically from the recurrence relation (1.2.49) in a stable fashion.

θ π %

REMARK 2.7. In the special case j ¢ 2 we observe the asymptotic behaviour n

s β π

j 2 j ∞

¥ n

n 1 α j 2

s j

& & λ

Decay rate of the convolution kernel. Since j ¢ 1 the relation (1.2.62) shows that s0 ,

¤ n

3 $ 2

¦ sJ ¢ O n , which agrees with the decay of the convolution kernel in the differential TBCs (1.1.8), (1.1.9). To show this property we consider the left TBC (1.1.9) and obtain after an integration by parts

t ¤

¤ ¦ d ψ 0 ¥ τ

ψ ! τ ¥ ¦

x 0 t ¢ c d

dt 0 t ¡ τ

¤ ¤ (1.2.65) ¤ ε

t t ¡

¦ ¥ ¦ ¥ ¦ ψ ¥ τ ψ τ ψ ε

d 0 1 0 0 t

¡ !

! τ τ

¤ £

¢ c d d

¡ $ τ ¡ τ 3 2 ε

dt t ε t 2 0 t ¦

with ¡

π 2 i 1 i

4 !

¢ (1.2.66) c ¢ e π π To compare the discrete convolution in the DTBCs with the continuous convolution in the dif- ε ∆ ferential TBCs we consider the following discretization (with ¢ t)

t ε ¤ n 1

ψ ¥ τ ¦

c 0 c 3 2

¡ ¡

¡ k

/ ¦ ¤ τ ψ ∆ ∆

(1.2.67) d ∑ 0 n k t t $ ¡ 3 2 2 0 t τ ¦ 2 k ) 1 If we compare this with (1.2.43a) written in the form

ψn ¡ ψn n 1

1 n k 0 ¡

1 0 k ¡ n 1 n n

ψ ψ ψ ψ ¥ ¥ £ (1.2.68) ¢ ∑ s s n 1 ∆x ∆x 0 0 1 0 0 0

k ) 1

we would roughly expect

c∆x i 1 ∆x $ n $

¡ 3 2 3 2

! ! !

¥ ∞

(1.2.69) s n ¢ n n

0 + 2 ∆t 2 π ∆t n In Figure 1.4 we compare the s0 for increasing time levels n with the r.h.s. of (1.2.62). The

∆ ∆ %

¢ ¢ used parameters are ¢ 1, t 10 , x 1 160 and the potential is set to zero. Figure 1.4

22 1. THE SCHRODINGER¨ EQUATION n

displays a quite good agreement of the asymptotic behaviour of the s0 with the predicted one n in (1.2.62). The values of s0 oscillate around the rate (1.2.69).

Asymptotic Behaviour of Convolution Coefficients s(n) 0 0

s(n) −1 0 s(n) asympt. 0 n−3/2 rate −2

−3

) −4 (n) 0 (s 10

log −5

−6

−7

−8

−9 1 1.5 2 2.5 3 3.5 4 log (n)

10 n FIGURE 1.4. New discrete TBC: convolution coefﬁcients s0 compared to the r.h.s. of (1.2.62) and the decaying rate (1.2.69).

2.4. Stability of the resulting Scheme. We shall now discuss the stability of the Crank– Nicolson ﬁnite difference scheme in connection with the DTBCs (1.2.43). Since the discrete whole–space solution satisﬁes the discrete TBCs (1.2.43), it is trivial that the implicit scheme (1.2.1), (1.2.43) for the IBVP can be solved at each time level n. To prove

unique solvability and stability of the scheme, a discrete analogue of (1.1.19) can be derived.

? §

¥ ¥ ¥

We sum up (1.2.2) for the ﬁnite interior range j ¢ 1 2 J 1, use the summation by parts rule:

J 1 J 1

¡ ¡

∆ @ ∆ £ (1.2.70) x ∑ g jD f j ¢ x ∑ f jD g j f g f g

x x J 1 J 0 0

) j ) 1 j 0

2 @

and obtain (note that Dx ¢ Dx Dx )

J 1 1 ∆ J 1 1 2 ∆ J 1 1 1 2

n i x n i x n n ¡ ∆ ψ 2 ψn ¡ ψ 2 2 ψ 2

x ∑ ¯ D ¢ ∑ D ∑ V

j t j 2 x j j j

) ) (1.2.71) j ) 1 j 0 j 1

1 1 1 1

i n n n n ψ 2 ψ 2 ¡ ψ 2 ψ 2

£ ¯ D ¯ D J x J 1 0 x 0

2 Finally, taking the real part using (1.2.5) yields

1 1 1 1

n n n n

n 2 2 2 ¡ 2 2

" @ ¥ " ψ ψ ψ ψ ψ

(1.2.72) D ¢ Re i ¯ D Re i ¯ D t 2 J x J 0 x 0 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 23

2 ψn 2 ∆ J 1 ψn 2 " & & " ∑ with the discrete L –norm deﬁned by : ¢ x . After summation with respect to 2 j ) 1 j the time index we get from (1.2.72): (1.2.73)

N 1 1 N 1 1

n n n n

N 1 2 0 2 2 2 ¡ 2 2

" " " @ ψ " ψ ∆ ψ ψ ψ ψ £ ¢ t Re i ∑ ¯ D Re i ∑ ¯ D

2 2 J x J 0 x 0 ) n ) 0 n 0

N 1 1 N 1 1

¤ ¤

0 0

∆ ∆

t n n n t n n n ¡

ψ0 2 ¡ ψ 2 ψ 2 ˜ ψ 2 ψ 2 ˜

" 1 ¦ 1 ¦ ¥ "

¢ Re i ∑ ¯ Re i ∑ ¯

2 ∆x J J J ∆x 0 0 0 )

n ) 0 n 0

0 0

n n

˜ ¡ δ0 ¥ ¢ where j : ¢ j n, j 0 J. Again, as in the continuous case, it remains to show that the boundary–memory–terms in (1.2.73) are of positive type. We concentrate on the boundary term

at j ¢ J and deﬁne the ﬁnite sequences

1 1

§ §

n n

ψ 2 ˜ n ψ 2 1 ¥ ¥ ¥ ¥ ¥ ¥

¢ ¢

(1.2.74) fn ¢ J J gn J n 0 1 N

N .

transformed DTBC (1.2.18) yields

¤ ¤ ¤

z £ 1 .

ˆ N ¡

¦ ¦ ν ¦ - ψ ¢ fn f z ¢ ˆ J z 2 z 1

(1.2.75) 2 ¤

iR ¤ ¡

ψN ¡ κ 2

¦ ¦ ¥

£ £ £ ¢ ˆ z z 1 i z 1 Az 2Bz C J

4 7

N ¤ N n n

& &A© ψ ¦ ψ

∑ where ˆ z ¢ z is analytic on z 0. The expression above in the curly brackets is

J n ) 0 J

& &=© & &B

analytic for z 1 and continuous for z 1, since the zeros z1 2 of the square root are given ¤

¤ ˆ ∞

6 % ¦ % & & ¦ *& &3

¢ ¢ by z1 2 B i4 R A with z1 2 1. Therefore f z is analytic on 1 z . Note that we

ν ¤ & & have to choose the sign in (1.2.75) such that it matches with 2 z ¦ for z sufﬁciently large. For

the second sequence gn we obtain

¤ ¤

z £ 1

. N

¦ ¦ ¥

- ψ ¢ (1.2.76) g ¢ gˆ z ˆ z

n 2 J

*& &3 i.e. gˆ z ¦ is analytic on 0 z ∞. Now the basic idea is to use Plancherel’s theorem in the form (Z.7) which gives

N 2π ¤ 1 2 ¤ 2

N ¡

& ¦ ¦ & ψ ν ϕ £ Re i ∑ f g¯ ¢ Re i z 1 ˆ z 2 z 1 d n n J iϕ 8π 0 z ) e (1.2.77) n ) 0

2π

1 ¤ ϕ

¡ ν i ϕ ¦

¢ Im 2 e d 8π 0

¤ 2

& & ν ¦ £ We remark that the pole of 2 z at z ¢ 1 is “cancelled” by z 1 . From (1.2.77) we conclude that the discrete L2–norm (1.2.73) is non–increasing in time if

ν ¤ iϕ ϕ π

¥ D ¨E ¥ F¥ (1.2.78) Im 2 e ¦C 0 0 2 holds. This property of ν2 can be shown in the following way. If we deﬁne

iR z ¡ 1

¡ κ £ (1.2.79) y ¢ i

2 z £ 1

then (1.2.21) simply reads

¤ ¤

ν ¦

£ £

(1.2.80) 2 y ¢ 1 y y 2 y

24 1. THE SCHRODINGER¨ EQUATION

¤ ¤ ϕ ¤

i ¡

¦ % ¦ % ¦

ϕ π ϕ

£ ¢ On the unit circle z ¢ e , 0 2 , we have z 1 z 1 itan 2 and therefore ϕ ¤ 2 ϕ

R Rκ ∆x ¦ 2 VL

¥ ϕ π

£ ¢ £ (1.2.81) y ¢ tan tan 0 2 2 2 2 ∆t 2

ν ¤ iϕ ϕ ϕ ϕ ϕ

is real. Consequently, 2 e ¦ becomes complex only in the interval a b, where a,

ϕ π ¥ b ¨G 0 2 solve ϕ ϕ ∆

a VL b t

¡ ¡

∆ ¥ ¥

£ ¢ (1.2.82) tan ¢ t tan V ∆ 2 L 2 2 x ¦ 2 2

and we have the requested result

¤ ¤ ϕ

i ¡

ν ¦ ¦H ¥ ϕ π £

(1.2.83) Im 2 e ¢ Im y 2 y 0 0 2

Imag ν (φ) 2 0.5

−0.5

−1

−1.5 0 π/2 π 3/2π 2π φ

ν ¤ iϕ ϕ π ¦

FIGURE 1.5. Imaginary part of 2 z on the unit circle z ¢ e , 0 2 .

9 4 ¢ The situation is illustrated in Figure 1.5, where we have set ¢ 1, VL 2 10 and used

∆ ∆ 9 5 ϕ ϕ ϕ %

¢ ¢

the parameters x ¢ 1 500, t 2 10 . This gives the following values for a, b: a

¤ ¤

¡ ¡

π ¦ / ϕ π ¦ /

¢ £ 2 £ 2atan 5 2 3 5216, b 2 2atan 0 2 5 8884. We then have the main result of this Section:

THEOREM 1.7 ( [T1]). The solution of the discretized Schroding¨ er equation (1.2.1) with the discrete TBCs (1.2.43) is uniformly bounded J 1

n 2 n 2 0 2

" ψ " ∆ ψ I" ψ " ¥ ¥ (1.2.84) 2 : ¢ x ∑ j 2 n 1

j ) 1 and the scheme is thus unconditionally stable.

REMARK 2.8. It can also be shown that (1.2.43) is a consistent discretization of the differential BCs (1.1.8), (1.1.9).

2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 25 n Simplified DTBC. The decay of the s j shown in (1.2.62) motivates to consider a simplified version of the DTBC (1.2.43) with the convolution coefficients cut off at an index M. This means that only the “recent past” (i.e. M time levels) is taken into account in the convolution in (1.2.43):

n 1

0 n k ¡

n ¡ n k n 1

¥ ψ ψ ψ ψ ¥ (1.2.85a) 1 s0 0 ¢ ∑ s0 0 1 n 1

k ) n M

n 1

0 n k ¡

n ¡ n k n 1

ψ ψ ψ ψ ¥

(1.2.85b) J 1 sJ J ¢ ∑ sJ J J 1 n 1

k ) n M This, of course, reduces the perfect accuracy of the DTBC (1.2.43), but it is numerically cheaper while still yielding reasonable results for moderate values of M. We remark that the numerical stability of the scheme with simpliﬁed DTBC depending on the value of M is not anymore obtained automatically. This issue is currently under investigation [D1].

2.5. Numerical Results. In this Section we present an example to compare the numerical results from using our new discrete TBC to the solution using other discretization strategies of the TBC for the Schrodinger¨ equation (1.1.1). We also show the numerical effect if the DTBC

is simpliﬁed by (1.2.85). Due to its construction, our DTBC yields exactly (up to round–off

¥ errors) the numerical whole–space solution restricted to the computational interval 0 L . The calculation with discretized TBCs requires the same numerical effort. However, the solution may (on coarse grids) strongly deviate from the numerical whole–space solution.

I ¤

ψ ¦

Example. This example shows a simulation of a right travelling Gaussian beam x ¢

¤ ¤ ¡

¡ 2 ¦§

exp i100x 30 x 0 5 ¦ at four consecutive time steps evolving under the free Schrodinger¨

∆ % ∆

¢ ¢ equation ( ¢ 1) with the rather coarse space discretization x 1 160 and the time step t 2 5

10 . Discretizing the analytic TBCs via (1.2.7) (scheme of Mayﬁeld [T16]) or as in Baskakov and Popov [T6] induces strong numerical reﬂections. Our discrete TBCs (1.2.43), however,

yield the smooth numerical solution to the whole–space problem, restricted to the computational

¥ interval 0 1 (up to round–off errors). We observe in Figure 1.6 the artiﬁcial reﬂections travelling to the left induced by discretizing

the analytic TBC while the solution with the new discrete TBC leave the computational domain

without any numerical reﬂections. At time t ¢ 0 01 the solution with the DTBC has almost

¥ completely left the domain 0 1 and the solutions with discretized TBCs contain a reﬂected wave packet with the maximum modulus (which corresponds to the maximum error) of around 0.17 for the approach of Mayﬁeld and around 0.025 in case of the discretized TBC of Baskakov and Popov.

Now we present the results when using the simpliﬁed DTBC (1.2.85) and want to compare

the outcome with the discretized TBCs at time t ¢ 0 01. All these boundary conditions need a comparable computational effort. The cut–off value M is chosen appropriately, such that the

simpliﬁed DTBC yields similar results with respect to the numerical reﬂections at the right 9§9?9

boundary x ¢ 1. As a reference we also plot the solution with the discrete TBCs ( ).

We see in Figure 1.7 that the solution with the simpliﬁed discrete TBCs (1.2.85) with M ¢ 5 is already better than the solution with the discretized analytic TBCs (1.2.7) from [T16]. We observe in Figure 1.8 that the error of the solution with the discretized analytic TBC

from [T6]. lies between the errors of the solutions with the simpliﬁed discrete TBCs (1.2.85) ¢ using the cut–off value M ¢ 30, M 35. 26 1. THE SCHRODINGER¨ EQUATION

Schroedinger: t=0.004 Schroedinger: t=0.006 1

1 new discrete TBC 0.9 new discrete TBC discretized TBC (Mayfield) discretized TBC (Mayfield) discretized TBC (Baskakov & Popov) discretized TBC (Baskakov & Popov) 0.8

0.8 0.7

0.6 0.6

| | 0.5 ψ ψ | |

0.4 0.4

0.3

0.2 0.2

0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x

Schroedinger: t=0.008 Schroedinger: t=0.01 0.25 0.18

new discrete TBC discretized TBC (Mayfield) new discrete TBC 0.16 discretized TBC (Baskakov & Popov) discretized TBC (Mayfield) discretized TBC (Baskakov & Popov) 0.2 0.14

0.12

0.15 0.1 | | ψ ψ | | 0.08 0.1

0.06

0.04 0.05

0.02

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x x

ψ ¤

& ¥ ¦(&

¢ ¢ ¢ FIGURE 1.6. Solution x t at time t ¢ 0 004, t 0 006, t 0 008, t 0 01: the solution with the new discrete TBCs (—) coincides with the whole– space solution, while the solution with the discretized analytic TBCs (1.2.7) from

[T16] ( ) or from [T6] ( 9§9§9 ) introduces strong numerical reﬂections. @J@J@

3. DTBC for non–compactly supported Initial Data

I ¤ In this Section we show how to drop assumption (A1), i.e. here the initial data ψ x ¦ need not be compactly supported inside the computational domain. We only assume that the initial I ¤ function ψ x ¦ is continuous. First we review the derivation of the TBC on the continuous level and mimick this derivation strategy afterwards for the discrete scheme.

3.1. The Transparent Boundary Condition. Here we review the derivation of the (continuous) TBC from [T13]. In the case of the free Schrodinger¨ equation with non–compactly supported initial data ψI the Laplace transformed (using (L.2)) right exterior problem (1.1.5) now reads

2 2 I ¤

ψ ¦ ¥ ¢

(1.3.1a) vˆxx £ c svˆ c x x L ¤

¤ ˆ ¦ ¦ ¥ ¥ Φ

(1.3.1b) vˆ L s ¢ s 3. DTBC FOR NON–COMPACTLY SUPPORTED INITIAL DATA 27

Schroedinger: t=0.01 0.25 discretized TBC (Mayfield) discrete TBC discrete TBC: M=5 discrete TBC: M=4 0.2

0.15 | ψ |

0.1

0.05

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

¥ ¦#& & ψ

FIGURE 1.7. Solution x t at time t ¢ 0 01: the solution with the dis-

cretized TBCs of Mayﬁeld [T16] (—) in comparison to the solution using the ¢

simpliﬁed DTBC (1.2.85) with M ¢ 4 ( ) and M 5 ( ).

@JK @JK @J@J@

Schroedinger: t=0.01 0.03

discretized TBC (Baskakov & Popov) discrete TBC discrete TBC: M=35 discrete TBC: M=30 0.025

0.02 |

ψ 0.015 |

0.01

0.005

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

& ψ ¥ ¦#&

FIGURE 1.8. Solution x t at time t ¢ 0 01: the solution with the dis-

cretized TBCs of Baskakov and Popov [T6] (—) in comparison to the solution ¢

using the simpliﬁed DTBC (1.2.85) with M ¢ 30 ( ) and M 35 ( ).

@JK @JK @J@J@

¦ % £ where we set c ¢ 1 i . Again, the idea is to solve this inhomogeneous second order

differential equation (1.3.1) explicitly. The homogeneous solution is

¤ ¤ ¤

ic s x L ic s x L

¥ ¦ ¦ ¦ ¥ © ¥

8 8 £ (1.3.2) vˆhom x s ¢ C1 s e C2 s e x L 28 1. THE SCHRODINGER¨ EQUATION and according to [M3, (14.31)] a particular solution of (1.3.1a) is given by

x x

¤ ¤ ¤

L L

ic s x x I ¡ ic s x x I

< < < <

ψ ψ

¥ ¦ ¦ ¦ ¥ © ¥

8 8 (1.3.3) vˆpar x s ¢ e x dx e x dx x L 2i s L L i.e. the general solution is

(1.3.4)

¤ ¤ ¤

¥ ¦ ¥ ¦ ¥ ¦ £ vˆ x s ¢ vˆhom x s vˆpar x s

x ¤ ¤ c

ic sL ic sxL I ic sx

< <

¦ ψ ¦

8 8 8 £ ¢ C1 s e e x dx e 2i s L

∞ ∞

¤ ¤

¤ c c

L L

ic sL ¡ ic sx I ic sx I ic sx

! !

< < < <

¦ ¦

¦ ψ ψ

8 8 8 8 £ £ C2 s e e x dx e x dx e 2i s L 2i s x ∞ We note that the last term in (1.3.4) is bounded for ﬁxed s and x . Since the solutions have

1 i

ic sx @ sx

∞

¢ 8 to decrease as x , the idea is to eliminate the growing factor e 8 e by simply choosing

∞

¤ ¤ c

ic s xL L I

< <

¦ ¦ ψ 8 (1.3.5) C2 s ¢ e x dx

2i s L ¤

Consequently, we obtain C1 s ¦ from the boundary condition (1.3.1b) :

∞

¤ ¤ ¤

c L

¡ ic s x L I

< <

¦ ¦ ¦ Φ ψ 8 (1.3.6) C1 s ¢ s e x dx 2i s L From this we get the following representation of the transformed right TBC :

2 ∞

¤ ¤ ¤

c L

¡ ic s x L I

< <

¥ ¦ ¦ ¦ 8 vˆx L s ¢ ic sC1 s e x dx 2 L

(1.3.7) ∞

¤ ¤

¡ 2 ic s x L I <

ˆ <

¦ ψ Φ ¦ 8 ¢ ic s s c e x dx L It remains to inverse transform (1.3.7). If we further assume that ψI is continuously differentiable, then integration by parts yields:

∞

¤ ¤ ¤ ¤

ic ic

L ¡

¡ I ic s x L I

! !

< <

¦ ¥ ¦ Φ ¦ ψ ¦ ψ 8 (1.3.8) vˆx L s ¢ s s L e x x dx s s L The inverse Laplace transformation using the convolution theorem gives

t ¤ ∞

¤ ¤

ψ ¥ τ ¦ ic L 1 t 1 L

¡ ic s x L I

! ! !

< <

ψ τ ψ

¥ ¦ M ¦ 8 (1.3.9) x L t ¢ d ic e x x dx π 0 t ¡ τ s L ψI Finally, if x is integrable for x © L, Levy proved ( [T13, Theorem 3.1]) that the integration and the inverse Laplace transform can be interchanged in (1.3.9) to obtain (with (IL.8)) the right TBC

¤ ∞ 2

t N

¤ ψ τ

¦ ¥ i x L

ic L ic

t @

¡ I

! !

ψ ! 2 t ¥ ¦ τ ψ ¦ (1.3.10) x L t ¢ d x x e dx π 0 t ¡ τ πt L

I ¤

REMARK 3.1. Clearly, if x ¢ 0 for x L then (1.3.10) reduces to the previously ob-

! π ¡

¡ i 4 tained right TBC (1.1.8) in the potential–free case (note that 2e ¢ i 1). As motivated in the previous Section we will not discretize this TBC. Instead we will show now how to derive the TBC on a fully discrete level by mimicking the derivation of the continuous TBC. 3. DTBC FOR NON–COMPACTLY SUPPORTED INITIAL DATA 29 3.2. The Discrete Transparent Boundary Condition. First we show how to solve an in-

homogeneous second order difference equation with constant coefﬁcients of the form

γ ¥

£ £ ¢

(1.3.11) Uj 1 aUj bUj 1 j j J 1 We already know from Section 2 that the two linearly independent homogeneous solutions take

α j β j αβ the form , , j J with ¢ b. A particular solution Vj of (1.3.11) can be found with the ansatz of “variation of constants” [E3], [E1] :

j 1 j 1

¥ α β £

(1.3.12) Vj 1 ¢ c j d j j J It follows that

j j j j ¡

α β α β ¥

£ ¢ £ (1.3.13) Vj ¢ c j 1 d j 1 c j d j j J 1

if we force the condition ¤

j ¤ j

∆ ¦ α ∆ β ¢

(1.3.14) c j £ d j 0

∆ ∆ ¡

to hold. Here denotes the usual forward difference operator, i.e. c j ¢ c j 1 c j. Analo-

gously, again assuming (1.3.14), we obtain for Vj 1

¤ ¤

α j 1 β j 1 ∆ α j 1 ∆ β j 1 ¦ ¦

¢ £ £ £

(1.3.15) Vj 1 c j d j c j d j

Inserting (1.3.12), (1.3.13), (1.3.15) into the difference equation (1.3.11) gives

¤ ¤

∆ α j 1 ∆ β j 1 γ

¦ ¦ ¥ ¢

(1.3.16) c j £ d j j together with the condition (1.3.14). This can easily be solved to obtain 1 1

j ¡ j

¥ ¥ ∆ α γ ∆ β γ ¢

(1.3.17) c j ¢ d j

j j ¡ α ¡ β α β i.e. we get the coefﬁcients

j 1 j 1

∆ 1 α mγ ¥ ¥

£ ¢ £

(1.3.18) c j ¢ cJ ∑ cm cJ ∑ j J

α ¡ β m ) m ) J m J

j 1 j 1

¡ m

∆ β γ ¥

£ ¢ (1.3.19) d j ¢ dJ ∑ dm dJ ∑ j J

α ¡ β ) m ) J m J Consequently, the particular solution reads

α j β j

¢ £ Vj c j 1 d j 1

(1.3.20) 1 j 1 j ¡

α mγ α j ¡ β mγ β j ¥ ¥

£ £

¢ cJ ∑ dJ ∑ j J 1 ¡

α ¡ β m α β m ) m ) J m J and the general solution of (1.3.11) is of the form

1 j j ¡

j j j m ¡ j m

¥ ¥ α β α γ β γ

£ £ (1.3.21) Uj ¢ c d ∑ m ∑ m j J 1

α ¡ β ) m ) J m J which is the discrete analogue to the solution formula (1.3.3) in the continuous case.

Now we use (1.3.21) to design a boundary condition at j ¢ J. For that purpose we as-

&J & &J© & α β αβ ¢ sume 1, 1 (recall that b ¢ 1 for the Crank–Nicolson scheme for solving the 30 1. THE SCHRODINGER¨ EQUATION Schrodinger¨ equation). Proceeding analogously to the continuous case we have to eliminate the growing factor β j by choosing d appropriately as ∞

1 m β γ (1.3.22) d ¢ ∑ m α ¡ β

m ) J We obtain from (1.3.21)

j ∞

1 1

m j j m ¡

α γ α β γ ¥

£ £

(1.3.23) Uj ¢ c ∑ m ∑ m j J 1 ¡

α ¡ β α β

) m J m ) j 1

The value of c can be expressed with UJ 1: J 1 ∞ UJ 1 β 1

¡ m

¥ β γ (1.3.24) c ¢ ∑ m

αJ 1 α α ¡ β

m ) J

and inserting this into (1.3.23) with j ¢ J: ∞

J 1 J m

α β γ £ (1.3.25) UJ ¢ c ∑ m α ¡ β

m ) J yields

α 1 ∞ ¡

¡ βJ m α γ

UJ ¢ UJ 1 1 ∑ m ¡

β α β

m ) J (1.3.26) ∞

¡ 1 m

¥ α β β γ

¢ U ∑

J 1 J m

m ) 0 or equivalently ∞

m m

β α γ £ (1.3.27) bU ¢ U ∑ b

J 1 J J m

m ) 0 Finally, we want to apply these results to the discretized Schrodinger¨ equation (1.2.13) and

0 ¡ ψ derive the DTBC at j ¢ J in the situation, when the initial data j does not vanish for j J 1.

In this case the –transformed right exterior Crank–Nicolson scheme reads:

¤ ¤

¤ z 1 z

¡ ¡

ψ ¡ κ ψ ψ ϕ ¦ ¦ ¦ ¥ ¥

£ ¢

(1.3.28) ˆ j 1 z 2 iR i ˆ j z £ ˆ j 1 z j j J 1

£ z £ 1 z 1 ϕ where the inhomogeneity j is given by

2 0 0 0

¡ ¡

ϕ ∆ ψ ψ κψ ¥ £ (1.3.29) j ¢ x j iR j R j j J 1 We can use (1.3.27) to obtain the transformed right DTBC :

∞

¤ ¤ ¤

¤ z m

¦ ¦ ¦ ¥

ψ ¦ ν ψ ν ϕ £ (1.3.30) ˆ z ¢ z ˆ z ∑ z

J 1 2 J 1 J m

z £ 1 m ) 0 where ν1, ν2 are the two solutions of the quadratic equation (1.2.17). ϕ

REMARK 3.2. Again, (1.3.30) reduces to the DTBC (1.2.18) for j ; 0.

3. DTBC FOR NON–COMPACTLY SUPPORTED INITIAL DATA 31

¤ ¤ ¤ ¤

n 1 m . n 1

- ¦ ¦ - ¦

¦ ν ν

¢ In order to formulate the DTBC we deﬁne pm : ¢ 1 z and set J : 2 z . Using (IZ.7) we obtain by inverse transforming (1.3.30)

n 1 ∞ n

¤ ¤

0 0

0 n k k

¡ ¡

n ¡ n k n n k

ψ ψ ψ ϕ ϕ

¦ ¦ ¥

£ £ (1.3.31) ¢ ∑ 1 ∑ ∑ 1 p n 1

J 1 J J J J J m J m

) )

k ) 0 m 1 k 0

0 n

Since the coefﬁcients J asymptotically alternate in time, this formulation can be improved

0 0

n n n 1

£ and shortened by regarding once more sJ : ¢ J J , which gives ﬁnally the DTBC for non–compactly supported initial data :

n 1 ∞

0 n k n ¡ n ¡ n k n 1

ψ ψ ψ ψ ϕ ¥ £ (1.3.32) s ¢ ∑ s ∑ p n 1

J 1 J J J J J 1 m J m

) k ) 0 m 1

REMARK 3.3. Note that in contrast to the DTBC in Section 2 the r.h.s. (1.3.32) for n ¢ 1 is not zero but

∞

1 ¡

ψ1 ¡ 0 ψ1 1 ψ0 ψ0 ϕ £ (1.3.33) ¢ s ∑ p

J 1 J J J 1 m J m

m ) 1 In practical situations the sum (over m) in (1.3.32) of course has to be ﬁnite (e.g. up to an

index m ¢ M). This means that the initial condition is still compactly supported, but possibly

§ §

¥ ¥ ¥ outside of the computational interval. The coefﬁcients pm , m ¢ 1 2 M, can be calculated recursively by “continued convolution”, i.e.

n n

n 1 . n n k k n n k k

- ¦ ¥ ¥ ¥

¢ ¢

(1.3.34) p1 ¢ 1 z p2 ∑ p1 p1 p3 ∑ p2 p1 etc. ) k ) 0 k 0

Since this computation is rather costly (even when using fast convolution algorithms with M n

FFTs [S5, Chapter 4]) we seek for another way to calculate ∑ p ϕ , n 1. The key idea )

¤ m 1 J m

is to use the quadratic equation (1.2.17) for ν1 z ¦ in order to construct a recurrence relation for

¤ n ν

the pm (w.r.t. m). Equation (1.2.17) for 1 z ¦ gives

¤ ¤

¤ iR z 1

νm 1 κ νm ¡ νm 1 ¦ ¦ ¦ £ 1 z ¢ 2 1 i 1 z 1 z

(1.3.35) 2 z £ 1

¤ ¤

¤ z ¡

m ¡ m m 1

¦ ¦ ¥ ν ¦ ν ν ¢ c1 1 z c2 1 z 1 z m 1

z £ 1

£ £ ¢ with c1 ¢ 2 iR R and c2 2iR. An inverse –transformation gives

n n n k n

¡ ¡

¡ k

¦ ¥ ¥

(1.3.36) p ¢ c pm c ∑ 1 pm p m 1

m 1 1 2 m 1

k ) 0

n δ0 n 1 ν .

- ¦

with the starting sequences p0 ¢ n and p1 1 z , n 0. To circumvent the convo-

n n n 1 1

£ ¢

lution in (1.3.36) we consider qm : ¢ pm pm , pm 0 and obtain

n n n n

¡ ¡

(1.3.37) q ¢ c qm c pm q m 1

m 1 1 2 m 1 to use in the DTBC of the form

n 1

0 n k n

¡ ¡

n ¡ n k n 1 n 2

¥ ψ ψ ψ ψ ψ ¥ £

(1.3.38) J 1 sJ J ¢ ∑ tJ J 2 J 1 J 1 SM n 1

k ) 0

32 1. THE SCHRODINGER¨ EQUATION

n n n 1

£ where tJ : ¢ sJ sJ , n 1 and

n n ϕ ¥

(1.3.39) S : ¢ ∑ q n 1 M m J m

m ) 1

REMARK 3.4. For n ¢ 1 (1.3.32) reads:

0 1 1 ¡

ψ1 ¡ ψ1 ψ0 ψ0 £

(1.3.40) J 1 sJ J ¢ tJ J 2 J 1 SM

The calculation of (1.3.39) with the aid of the recursion formula (1.3.37) is done by the

following algorithm

n δ0 δ1 £

1. q0 ¢ n n

n n n 1 n

£ ¢

2. q1 ¢ p1 p1 s˜J

n n ϕ

3. S ¢ q

1 1 J 1

§ ?

¥ ¥

4. for m ¢ 1 M 1 do

n n n n

¡ ¡

q ¢ c qm c pm q

m 1 1 2 m 1

n n n ϕ £

S ¢ S q

m 1 m 1 J m 1

0 0

p ¢ q

m 1 m 1

§ ?

¥ ¥

for n ¢ 1 N do

n n n 1

p ¢ q p

m 1 m 1 m 1 end

end Here N denotes the maximum time index and s˜ n the summed coefﬁcients but with the other

J ¤ 9 sign. The computational effort of the above implementation of the DTBC is O M N ¦ , i.e. the same effort as when enlarging the computational domain sufficiently. The usage of the DTBC is especially beneficial when one needs several computations with the same initial data. Then the calculation of the additional term has only to be done once. The same applies when the initial field is concentrated far outside the computational domain. This is the case in radiowave propagation when computing coverage diagrams of airborne antennas. Alternatively, a second possible implementation is to consider the transformed DTBC (1.3.30) and to calculate numerically the inverse –transform of the finite sum once:

1 z m

ν ¦ ϕ

(1.3.41) F ¢ ∑ z n 1 J m

z £ 1 m ) 0 The DTBC then reads

n 1

0 0

0 n k

n ¡ n k

ψ ψ ψ ¥ £

(1.3.42) J 1 J J ¢ ∑ J J Fn n 1

k ) 0 The numerical inverse –transformation will be the topic of the next section.

REMARK 3.5. While the DTBC (1.3.32) solves the problem of initial data that are supported outside of the computational domain, the resulting numerical effort of this approach is 4. NUMERICAL INVERSE P –TRANSFORMATIONS 33

not completely settled yet and subject to further investigations. In particular one has to com-

m m pare an “optimal” computation algorithm for the coefﬁcients pn or qn with simulations on a sufﬁciently enlarged computational domain. 3.3. Numerical Results. Here we present the numerical results when using our new dis-

crete TBC (1.3.32) for the Schrodinger¨ equation (1.1.1). We use the same initial data as in Section 2.5, but shifted such that it is partially outside the computational domain 0 x L.

Again, due to its construction, our DTBC yields exactly (up to round–off errors) the numerical

¥ whole–space solution restricted to the computational interval 0 L .

I ¤

¦ ψ

Example. This example shows a simulation of a right travelling Gaussian beam x ¢

¤ ¤ ¡

¡ 2 ¦?

exp i100x 30 x 0 8 ¦ at three consecutive times evolving under the free Schrodinger¨ equa-

tion ( ¢ 1) with the rather coarse discretization of 161 grid points for the interval 0 x 1

∆ ∆ 9 5

¢ (i.e. x ¢ 1 160) and the time step t 2 10 . For the right exterior (computational) domain

we choose the same space step ∆x and use 60 grid points which results in the exterior interval

1 x 1 38125.

In the following Figure 1.9 we plotted the absolute value of the initial data and the solution

¢ ¢ obtained with the discrete TBCs (1.3.32) at the time steps t ¢ 0 002, t 0 004, t 0 006. One clearly sees in Figure 1.9 that the solution is solely propagated to the right and no artiﬁcial reﬂections are caused.

In this example the computation using the inhomogeneous DTBCs (1.3.32) needs approx-

imately the same CPU–time than just enlarging the domain to the interval 0 x 1 8 using a

simple Neumann boundary condition at x ¢ 0 and x 1 8. From Figure 1.9 one can guess that

the solution at t ¢ 0 004 has already reached the right boundary at x 1 8. Hence it is worth-

while in this example to use the inhomogeneous DTBCs whenever the solution for t © 0 004 is needed.

4. Numerical Inverse –Transformations The crucial point in the derivation of the DTBC in Section 2 was to ﬁnd the exact inverse –transformations. If it is not possible to calculate the convolution coefﬁcients analytically then the inverse –transformation can be performed numerically.

The numerical inversion of the –transformation is based on the simple observation that the

§ ?

- ¥ ¥

–transformation of the sequence fn , n ¢ 0 1 .

∞

. ˆ n 1

- ¦ ¥ ¨ ¥ & &3© ¥ ¢ (1.4.1) fn f z : ¢ ∑ fn z z CI z R

n ) 0

% is nothing else but a Taylor series in z˜ ¢ 1 z, i.e. the problem of calculating the inverse – ˆ ¤

transformation of f z ¦ is the numerical evaluation of the Taylor coefﬁcients of the function ¤

˜ ¤ ˆ ¦ % ¦ f z˜ : ¢ f 1 z . For that purpose we used here the FORTRAN subroutine ENTCAF (Evaluation of Normalized Taylor Coefﬁcients of an Analytic Function) from Lyness and Sande [C5]. n First we want to outline the method. The (normalized) Taylor coefﬁcients r fn can be obtained by Cauchy’s integral representation :

¤ r

n ˜ n 1

¦ ¥ ¥ (1.4.2) r fn ¢ π f z˜ z˜ dz˜ r R 2 i Q

where R denotes a circle around the origin with radius r smaller than the radius of convergence R of the Taylor series. The approximation to fn based on using an N–point trapezoidal rule for 34 1. THE SCHRODINGER¨ EQUATION

Schroedinger: t=0 Schroedinger: t=0.002 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

| 0.5 | 0.5 ψ ψ | |

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x

Schroedinger: t=0.004 Schroedinger: t=0.006 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

| 0.5 | 0.5 ψ ψ | |

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x x

ψ ¤ & ¥ ¦(&

¢ ¢ ¢ FIGURE 1.9. Solution x t at time t ¢ 0, t 0 002, t 0 004, t 0 006: the solution with the new discrete TBCs (1.3.32) coincides with the whole–space

solution and does not introduce any numerical reﬂections. N the contour integral is fn given by

N 1

π π § ?

N 1 2 k 2 k

n in N i N ¡

¥ ¥ ¥ ¥ ¢ (1.4.3) r f ¢ ∑ e f re n 0 1 N 1 n N

k ) 0

n N N

The approximation r fn is obtained by an iterative process. First approximations f , N ¢

§ § 0

¥ ¥ ¥

1 ¥ 2 4 8 are computed using (1.4.3). The convergence criterion is based on the knowledge

N ˜

∞ ¢ ¢ of the exact value of the limit of the sequence: limN : f0 f0 f 0 (cf. the Initial Value

Theorem Z.6 in the Appendix). After converging the second part consists in evaluating (1.4.3)

§ ?

¥ ¥ ¥ for n ¢ 0 1 N 1 using the stored function values obtained in the ﬁrst part. Since N is a power of 2 it is particularly appropriate to use a fast Fourier transform technique for this part. The user has to specify the required absolute accuracy εreq and the radius of computation r (the only restriction is that r must be less than R). ENTCAF returns an accuracy estimate εest

4. NUMERICAL INVERSE P –TRANSFORMATIONS 35 n N

together with approximations r fn and a number N, which are supposed to satisfy

§ §

N ¡

n ¡ n

¥ ¥ ¥ ¥ ¥ ¥ ε (1.4.4a) r fn r fn est n ¢ 0 1 2 N 1

n ε ? § 4 & & ¥ ¥ ¥ £

(1.4.4b) r fn est n ¢ N N 1 n N We see from (1.4.4a) that this algorithm naturally delivers approximations r fn with a uniform bound on the discretization error. An output status parameter indicates to the user whether or not convergence or roundoff errors have occurred. Exploiting the information of this output

parameter one could construct a driver program which ﬁnds the appropriate value of r by itself.

0 n

Due to the asymptotic behaviour (1.2.36) it is not advisable to calculate the j . Instead

n n we show how to compute the summed convolution coefﬁcients s j numerically. The s j were

deﬁned by:

0 0 0

n n n 1 0 0

¥ ¥ ¥ ¥

£ ¢ ¢

(1.4.5) s j : ¢ j j n 1 s j j j 0 J

We concentrate on the right BC at j ¢ J. If we assume l j 0 we have:

¤ ¤ ¤ ¤

n . 1

ν ¦ ¦ ¦ ¦ ¥ - ν ν

¢ ¢ £

(1.4.6) sJ 2 z £ z 2 z 1 z˜ ˜ 2 z˜

¤ ¤ ¤

¦ ¦ ν ¦ ν ν with ˜ 2 z˜ ¢ 2 z and 2 z given by formula (1.2.21).

4.1. Numerical Results. Here we present the numerical results when using the subroutine

n n ε ENTCAF to compute the convolution coefﬁcients J , sJ . In each example we chose req ¢

6 ε 15 10 and set the machine accuracy parameter M to 10 .

9 4 Example 1. The value for the potential VL was set to 2 10 and the discretization parameter

9 5

∆ % ∆

were taken from the example of Subsection 2.5, i.e. x ¢ 1 160, t 2 10 . We used the

computational radius r ¢ 0 92. For that parameter choice ENTCAF returned a number of N

ε 9

256 nontrivial calculated coefﬁcients and an estimated uniform absolute accuracy est ¢ 9 8302

7 n n

10 in case of the coefﬁcients J . For the summed coefﬁcients sJ we obtained N ¢ 128 and

ε 9 7 est ¢ 4 7684 10 . In the following Figure 1.10 we present the real and imaginary part of the numerically obtained coefﬁcients in comparison to the exact values.

(n) Re l (n) Im l J J 0.3 20

15 0.2

0.1 5

0 0

−5 −0.1

−10

−0.2 −15

−0.3 −20 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180

n n

0 0

n n FIGURE 1.10. Example 1: Values for Re J , Im J 36 1. THE SCHRODINGER¨ EQUATION

One serious problem is the rescaling of the computed coefﬁcients. Since the algorithm n N

yields approximations r fn with a uniform accuracy (1.4.4a) the computation is only reliable

N n N

to a limited number of n when calculating fn from r fn for r 1. Therefore some visible

0 0

n n numerical errors occur in the calculation of Re J for n 130 and Im J for n 170. On the left side of Figure 1.10 we observe the alternating behaviour shown in (1.2.36)

¤ 2

¤ ¤ ¤

0 ∆

n 8 x ¦ 125

¡ ¡

¡ n n n

¦ ¦ ¦ ¢ (1.4.7) 2iR 1 ¢ i 1 i 1 j + ∆t 8

Re s(n) Im s (n) J J 0.5 0.8

0.4 0.7

0.3 0.6

0.2 0.5

0.1 0.4

0 0.3

−0.1 0.2

−0.2 0.1

−0.3 0

−0.4 −0.1

−0.5 −0.2 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90

n n

n n FIGURE 1.11. Example 1: Values for Re sJ , Im sJ .

(n) (n) (n) (n) −4 Error in Re l , Re s −3 Error in Im l , Im s x 10 J J x 10 J J 2 (n) Error in Re l (n) J Error in Im l (n) J Error in Re s (n) J Error in Im s J 3

0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 n n

FIGURE 1.12. Example 1: Difference between the numerical and exact value

n n 9§9§9 of the real and imaginary part of J (—) and sJ ( ).

Example 2. In this second example we set the potential to zero and changed the compu-

∆ ∆ 9 4 %

¢ ¢

tational radius to r ¢ 0 95 and the step sizes x 1 1600, t 2 10 . For the coefﬁcients

0 n

J ENTCAF returned a number of N ¢ 256 nontrivial calculated coefﬁcients and an estimated

ε 9 6 n uniform absolute accuracy est ¢ 2 9802 10 . In case of the summed coefﬁcients sJ we

4. NUMERICAL INVERSE P –TRANSFORMATIONS 37

ε 9 7

¢ obtained N ¢ 128 and est 1 2288 10 . As before we show the real and imaginary part of the numerically obtained coefﬁcients in comparison to the exact values. Again, on the left side

Re l (n) Im l (n) J J 0.04 0.08

0.03 0.06

0.02

0.04 0.01

0 0.02

−0.01 0

−0.02

−0.02 −0.03

−0.04 −0.04 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180

n n

FIGURE 1.13. Example 2: Inverse –transformation using ENTCAF:

0 0

n n

Numerically obtained values for Re J , Im J compared to the exact ones.

0 n of Figure 1.13 one can see the alternating behaviour of the coefﬁcients j and the numerical

errors due to the rescaling of the computed coefﬁcients for approximately n 30.

(n) (n) Im s Re s J J 0.01 0.02

0.008 0.018

0.006 0.016

0.004 0.014

0.002 0.012

0 0.01

−0.002 0.008

−0.004 0.006

−0.006 0.004

−0.008 0.002

−0.01 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90

n n

FIGURE 1.14. Example 2: Inverse –transformation using ENTCAF:

n n Numerically obtained values for Re sJ , Im sJ compared to the exact ones.

Example 3. Finally we use the settings of the second example and intend to use a compu-

0 n

tational radius to r ¢ 1 to circumvent the problem of the rescaling. For the coefﬁcients this

¤ J ν ¦ cannot be done due to the singularity of 2 z at z ¢ 1. On the other hand, this singularity is 38 1. THE SCHRODINGER¨ EQUATION

(n) (n) (n) (n) −6 Error in Re l , Re s −6 Error in Im l , Im s x 10 J J x 10 J J 1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90

n n

FIGURE 1.15. Example 2: Inverse –transformation using ENTCAF:

Difference between the numerical and exact value of the real and imaginary part

n n 9§9§9 of J (—) and sJ ( ).

removed in (1.4.6):

¤ ¤

n .

¦ ¦ - ν £ sJ ¢ 1 z˜ ˜ 2 z˜ (1.4.8)

iR iR 4i ¤

¡ ¡

¦ ¥

¢ £ £ 1 z˜ y 1 z˜ £ y y

2 2 R

¤ ¤

¡ κ ¦§¦

£ £

with the abbreviation y ¢ 1 z˜ i z˜ 1 . n

For the summed coefﬁcients sJ ENTCAF returned a number of N ¢ 1024 nontrivial calcu-

ε 9 6

lated coefﬁcients and an estimated uniform absolute accuracy est ¢ 2 2492 10 . The maximal

n 9 6 6 n absolute error for Re sJ is 2 8609 10 and 2 1455 10 for calculating Im sJ . 4. NUMERICAL INVERSE P –TRANSFORMATIONS 39

(n) (n) −3 Re s −4 Im s x 10 J x 10 J 1 10

0 5

−1 0 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800

n n

FIGURE 1.16. Example 3: Inverse –transformation using ENTCAF:

n n Numerically obtained values for Re sJ , Im sJ compared to the exact ones. 40 1. THE SCHRODINGER¨ EQUATION CHAPTER 2

The Convection–Diffusion Equation

Here we will demonstrate how the derivation of the DTBC of the preceding Chapter carries over to the case of a linear convection–diffusion equation. This type of parabolic equation arises for example in the study of heat conduction, reaction diffusion problems and when linearizing

the Navier–Stokes equation. The results will be generalized to the θ–scheme with the implicit- ness parameter 0 θ 1 and to a ﬂexible discretization of the convection term. A short version of this Chapter can be found in [P2]. This approach can be generalized to a vector–valued parabolic equation describing (second order) ﬂuid stochastic Petri nets [P7].

1. Transparent Boundary Conditions In this Section we shortly present the derivation of the (analytic) TBC. For simplicity of the notation we restrict ourselves here to a constant coefﬁcient parabolic equation in one space

dimension:

¡ ¡

¥ ¨ ¥ © ¥

ut ¢ auxx bux cu x IR t 0

(2.1.1) ¤ I

¥ ¦ ¦

u x 0 ¢ u x Of course, in general the coefﬁcients in (2.1.1) may depend on x and t but for our purposes we shall assume that they remain constant outside of the computational domain. Without loss of

generality we assume for the reaction rate c 0 (otherwise this property can be achieved by a

¡ ¦

change of variables v ¢ uexp c˜t ). ¢ 1.1. Derivation of the TBC. Here we determine the TBCs at x ¢ 0 and x L, such that

the solution of the resulting IBVP coincides with the solution of the whole–space problem ¤

¤ I

¦ ¦

(2.1.1) restricted to 0 ¥ L . Once more we have to assume that the initial data u x is compactly

¤ ¦ supported in the computational domain 0 ¥ L . Proceeding analogously to the formal derivation

of the TBCs in Chapter 1 we consider the interior problem

¡ ¡

¥ ¥ © ¥

ut ¢ auxx bux cu 0 x L t 0 ¤

¤ I

¥ ¦ ¦ ¥

u x 0 ¢ u x

¤ ¤

(2.1.2) ¤

¥ ¦ ¦ ¥ ¦ ¥

ux 0 t ¢ T0u 0 t

¤ ¤ ¤

¥ ¦ ¦ ¥ ¦ ¥

ux L t ¢ TLu L t

and obtain the Dirichlet–to–Neumann map T0 by solving the left exterior problem:

¡ ¡

¥ ¥ © ¥

vt ¢ avxx bvx cv x 0 t 0

¥ ¦ ¥

v x 0 ¢ 0

¤ ¤ ¤

¥ ¦ Φ ¦ ¥ © ¥ Φ ¦ ¥ ¢

(2.1.3) v 0 t ¢ t t 0 0 0

¦ ¥ ∞ ¥

v t ¢ 0

¤ ¤ ¤

¦ ¥ ¦ Φ ¦ T0 t ¢ vx 0 t

41 42 2. THE CONVECTION–DIFFUSION EQUATION

(2.1.3) is coupled with (2.1.1) by the assumption that u, ux are continuous across the artiﬁcial boundary at x ¢ 0. Since the initial data vanishes for x 0, we can solve (2.1.3) explicitly by

the Laplace–method and obtain the transformed left exterior problem:

¡ ¡

¦ ¥ ¥ ¢

avˆxx bvˆx c £ s vˆ 0 x 0

(2.1.4) ¤

¦ ¦

¥ Φˆ

vˆ 0 s ¢ s

¤ ¤ λ

¡ ∞ Φˆ 1 s x

¥ ¦ ¦

The solution which decays as x is simply vˆ x s ¢ s e , x 0, with !

¤ b 1

6 ¥

λ ¦ η

¢ £ (2.1.5) 1 2 s s 2a a and the parameter 2

η £ (2.1.6) ¢ c 0 4a

Consequently, the transformed left TBC reads:

¤ ¤ ¤

¥ ¦ λ ¦ ¥ ¦

(2.1.7) uˆx 0 s ¢ 1 s uˆ 0 s We remind the reader that in (2.1.5) ! denotes the branch of the square root with nonnegative real part. After the inverse Laplace transformation the left TBC reads:

t ¤ ητ

¤ τ ¦

b 1 ηt d u 0 ¥ e

! !

¦ ¥ ¦ ¥ τ £ (2.1.8) ux 0 t ¢ u 0 t e d 2a aπ dt 0 t ¡ τ We observe again that (2.1.8) is nonlocal in t (of memory–type), i.e. the computation of the

solution at some time uses the solution at all previous times. The right TBC at x ¢ L is derived similarly:

t ¤ ητ

¤ τ ¦ b 1 η d u L ¥ e

¡ t

! !

¦ ¥ ¦ ¥ τ

(2.1.9) ux L t ¢ u L t e d 2a aπ dt 0 t ¡ τ

REMARK 1.1 (Inhomogeneous Equation). We note that it is possible to consider (2.1.1)

¤ ¤ ¤

¥ ¦ ¥ ¦ ¥ ¦ with an inhomogeneity f x t provided that f x t ¢ f is constant outside of 0 L . In this

case the transformed left exterior problem is an inhomogeneous ODE in x:

¡ ¡

¡ 1

¦ ¥ ¥ ¢

avˆxx bvˆx c £ s vˆ s f x 0

¤ (2.1.10) ¤

Φˆ ¥ ¦ ¦

vˆ 0 s ¢ s and a (constant in x) particular solution reads

part ¤ f

¥ ¦ ¥ ¥

(2.1.11) vˆ x s ¢ x 0 ¦

s c £ s

i.e. the general solution to (2.1.10) is

¤ ¤ f λ f

¡ 1 s x ¤

ˆ ¤

¥ ¦ ¦ ¥ Φ ¥ £

(2.1.12) vˆ x s ¢ s e x 0

¦ ¦ £

s c £ s s c s ¤

with λ1 s ¦ given in (2.1.5). Therefore the transformed left TBC is given by:

¤ ¤ ¤

¤ f 1 1 ¡

ˆ ¡ ¦ ¦ ¦ ¦ ¥ ¥ λ Φ λ

(2.1.13) vˆx 0 s ¢ 1 s s 1 s

c s c £ s

and after the inverse Laplace transformation the left TBC reads: ¤ ¤ ητ

t ¡

¤ ¤ ¤

¦ ψ ¦ b 1 η d u 0 ¥ t t e

¡ t

! !

ψ τ

¥ ¦ ¥ ¦ ¦ ¥ £ (2.1.14) ux 0 t ¢ u 0 t t e d 2a aπ dt 0 t ¡ τ

1. TRANSPARENT BOUNDARY CONDITIONS 43 ¤

where ψ t ¦ is obtained using (IL.3):

¤ ¤ f

¡ ct

¦ ¦

(2.1.15) t ¢ 1 e c 1.2. Well–posedness of the IBVP. It is known that the IVP (2.1.1) is well–posed:

THEOREM 2.1 (Theorem 6.2.1, [F8]). The initial value problem for the equation (2.1.1) is

¥ ¦ well–posed, i.e. for any time T 0 there is a constant CT such that any solution u x t satisﬁes

t ¤

¤ I

¥ ¦(" " ¥ ¦(" " "

" τ τ

(2.1.16) u t L2 IR £ ux L2 IR d CT u L2 IR

0 for 0 t T. However, the well–posedness of the associated IBVP is not clear a-priori. While the exis-

tence of a solution to the 1D parabolic equation (2.1.2) with the TBCs (2.1.9), (2.1.8) at x ¢ 0

and x ¢ L is clear from the used construction it remains to prove the uniqueness of the solution.

¤ ¦ A straight forward calculation using the energy method, i.e. multiplying (2.1.1) with u x ¥ t and integrating by parts in x, yields

¤S ¤ ¤ ¤

1 d ¤

¡ ¡

2 2 ¡ 2

" ¥ ¦(" " ¥ ¦(" ¥ ¦ ¥ ¦ " ¥ ¦#"

u t 2 a u t 2 b u x t ux x t dx c u t 2

L 0 L L 0 L L 0 L

2 dt 0

¤ ¤

(2.1.17) x ) L

¥ ¦ ¥ ¦(&

£ au x t u x t

x x ) 0

¤ ¤

x ) L

¥ ¦ ¥ ¦(& ¥ au x t u x t x x ) 0

2 ¤

% ¦

if c b 4a (which can always be achieved by a change of variables). Finally, integrating

¤ ¦

in time using Plancherel’s Theorem for the Laplace transformation (Theorem L.3) (u 0 ¥ t and

¦ © u L ¥ t are extended by 0 for t T) gives (2.1.18)

¤ ¤ ¤ ¤

¤ I

" ¥ ¦(" I" " ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¦§

u t 2 u 2 £ 2a u L t u L t u 0 t u 0 t dt L 0 L L 0 L x x

¤ ¤ ¤

I ¤ 2 2

" ¦'& ¥ ¦(& ¦'& ¥ ¦(&

" λ ξ ξ λ ξ ξ ξ

¢ u 2 £ 2a i uˆ L i i u 0 i d L 0 L 2 1 IR

∞

¤ ¤ ¤ I ¤ 2 2

λ ξ ξ ¡ λ ξ ξ ξ " " ¦'& ¥ ¦(& ¦'& ¥ ¦(& ¥

£ ¢ u 2 4a Re i uˆ L i Re i u 0 i d L 0 L 2 1

¤ ¤ ¤ ¤ ¤ ¤

¡ ¡

¡ ¯ ¯ ¯

¦ ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¦ ¥ ξ ξ ξ ξ λ ξ λ ξ

¢ ¢ since uˆ 0 i ¢ uˆ 0 i , uˆ L i uˆ L i , 1 2 L i 1 2 L i . Now it can easily be checked that the conditions for well–posedness

λ ¤ ξ ξ

¥ ¨ ¥

(2.1.19a) Re 1 i ¦ 0 for IR

¥ ¨ (2.1.19b) Re λ2 iξ ¦ 0 for ξ IR λ

( 1 2 given by (2.1.5)) are fulﬁlled and we can state the following theorem.

THEOREM 2.2. The resulting parabolic IBVP is well–posed

¤ I

" ¦#" *" " ¥ © ¥ ¥

(2.1.20) u t 2 u 2 t 0 L 0 L L 0 L and this implies uniqueness of the solution to the parabolic IBVP. 44 2. THE CONVECTION–DIFFUSION EQUATION 2. Discrete Transparent Boundary Conditions Next we shall address the question how to adequately discretize the analytic TBC (2.1.8) for a chosen full discretization of (2.1.1). Instead of discretizing the analytic TBC (2.1.8) with its singularity we derive the discrete TBCs of the fully discretized problem based on the so–called

θ θ ∆ ∆ ¥ ¢ weighted average or –scheme for 0 1. With the uniform grid points x j ¢ j x tn n t

n ¤

¥ ¦ θ

and the approximation u j / u x j tn the –scheme for solving (2.1.1) reads:

2 θ σ θ θ ¡

n n ¡ n n ¥

(2.2.1) D u ¢ aD u bD u cu

t j x j x j j

with

n n 1 ¡ n

¥ θ θ ¦ θ £ (2.2.2) u j ¢ u j 1 u j 0 1 REMARK 2.1. While a uniform grid in x is necessary in the exterior domain, the interior grid may be nonuniform in x.

2 In the scheme (2.2.1) Dt denotes the usual forward and D the second order difference x quotient:

θ θ θ

n 1 ¡ n n n n

u u u 2u £ u θ

n j j 2 n j 1 j j 1

¥ ¢ (2.2.3) D u ¢ D u t j ∆ x j ∆ 2 t x ¦

σ ¤

σ ¡ £

The convection term is discretized by the parametrized difference quotient Dx ¢ Dx 1

σ ¦

Dx@ , i.e.

¤ ¤

¡ ¡

n ¡ n n

¦ σ σ ¦ σ

u £ 1 2 u 1 u

σ n j 1 j j 1

¥ ¥ σ

(2.2.4) D u ¢ 0 1

x j ∆t σ % which allows to switch between a centered difference quotient ( ¢ 1 2) or a one–sided differ-

σ σ ¢ ence quotient ( ¢ 0 or 1).

REMARK 2.2 (“upwind differencing”). One typical property of the parabolic equation (2.1.1)

is that

¤ ¤

< <

¥ ¦(& ¥ ¦ ¥ ©

sup & u x t sup u x t if t t ,

x , IR x IR

¥ ¦(& That is, the maximum value of & u x t will not increase as t increases (“maximum principle”). In [D10, Chapter 4.1.6] it was proven that the difference scheme (2.2.1) will have a similar property if and only if the two conditions

b∆x ¤ b∆x

¡ ¡

σ ¦ (2.2.5) σ 1 and 1 1

a a

σ %

are satisﬁed. We note that for a reasonable choice of σ (”upwind differencing”), i.e. 0 1 2

% for b 0 and 1 2 σ 1 for b 0 these conditions can always be fulﬁlled.

REMARK 2.3 (stability of the scheme). Secondly one can easily show using the von–Neu- mann analysis [F8] that the difference scheme (2.2.1) for solving the initial value problem (2.1.1) is stable (in the discrete L2–norm) provided that the stability condition

¤ ∆t

¤ (2.2.6) 1 2θ ¦ a 1 ∆ 2 x ¦

θ % holds and upwind differencing is used. I.e. for a rather implicit scheme ( 1 2) no additional condition has to be imposed to ensure stability. 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 45 2.1. Discretization strategies for the TBC. Here we want to compare three strategies to discretize the TBC (2.1.8) which is a rather delicate question. First we review two known discretization techniques from Mayﬁeld [T16] and Halpern [P3].

Discretized TBC of Mayfield. To compare our results we first repeat the ad-hoc discretization strategy of Mayfield applied to the convection–diffusion equation. According to the approach of Mayfield [T16] for the Schrodinger¨ equation one way to discretize the analytic TBC

(2.1.8) at x ¢ 0 in the equivalent form

¤ ¤

t η t τ t η t τ

¦ ¥ ¦ ¥ τ τ

a ux 0 e b u 0 e

! !

! τ τ ¥ ¦ ¥

(2.2.7) u 0 t ¢ d d ¡ π 0 t ¡ τ 2 aπ 0 t τ is for the ﬁrst integral

¤ ητ t ¡ n 1 t

¤ m 1

¦ ¥ τ τ ux 0 t e 1 η ∆ d

n m ¡ n m m t

! !

τ ¦ d / ∑ u u e τ ∆ 1 0 τ 0 x tm

m ) 0 ! (2.2.8) ¤ η ∆

n 1 n m ¡ n m m t

∆ ¦

2 t u1 u0 e

! !

¢ ∑

∆x £ m £ 1 m m ) 0 Discretizing the second integral in (2.2.7) analogously leads to the following discretized TBC for the convection–diffusion equation:

! n 1 n 1

¤ 0

π∆ 0

b x b

¡ ¡ n ¡ n n n m n m n m

! ˜ ˜

¦ ¥

¢ £ (2.2.9) u 1 £ u u ∑ u u m ∑ u m

1 2a 0 2 a∆t 0 1 0 2a 0 ) m ) 1 m 1 with the convolution coefﬁcients given by

ηm∆t 0 e

˜ !

¥ !

m ¢ £ m £ 1 m and η deﬁned by (2.1.6). On the fully discrete level (2.2.9) is not perfectly transparent any more and may possibly lead to an unstable numerical scheme as shown in Theorem 1.2 in case of the Schrodinger¨ equation.

REMARK 2.4 (Alternative Discretization of the TBC). In the context of underwater acous-

tics Thomson and Mayﬁeld [T26] proposed an alternative discretization of (2.1.8) to handle the ¤

η ¡ τ exponential terms more accurately when t ¦ is large. This approach will be presented in Chapter 3. Artificial BC of Halpern. Secondly we present the approach of Halpern [P3] which was generalized by Loheac´ in [P5], [P6] to the case that the diffusion coefficient a in (2.1.1) can depend on x. Halpern and Rauch derived in [P4] absorbing boundary conditions with variable coefficients, curved artificial boundary and arbitrary convection. The numerical study of this conditions were carried out in [P1] by Dubach. In [P3] Halpern developed a family of artificial boundary conditions for the linear convection– diffusion equation with small diffusion a. To start with we rewrite the transformed left TBC

(2.1.7) as

¤ ¤

¤ 1 2

¥ ¦ ¦ ¥ ¦

¢ £ £ (2.2.10) uˆx 0 s b £ b 4 c s a uˆ 0 s 2a 46 2. THE CONVECTION–DIFFUSION EQUATION Now Halpern’s approach consists of using Taylor or Pade´ approximations of the term in paren- theses in (2.2.10) with respect to a small value of a in order to obtain a local in t boundary

condition. A ﬁrst order Taylor approximation gives

¤ ¦

2 2a c £ s

¦ / & ¥ &

£ £ £ £

(2.2.11) b £ b 4 c s a b b

& & b

which leads to the transformed boundary condition

¤ ¤

& & £

b £ b c s

¥ ¦ ¥ ¦ ¢

(2.2.12) uˆx 0 s £ uˆ 0 s & 2a & b Finally an inverse Laplace transformation yields (depending on the sign of b) the ﬁrst order artiﬁcial boundary conditions:

¤ ¤ ¤

¤ b

£ ¢ (2.2.13a) inﬂow BC b 0 : u 0 t bu 0 t £ c u 0 t 0

t x a

¤ ¤ ¤ ¤

¦ ¥ ¦ ¥ ¦ ¥ ¦

£ ¢ (2.2.13b) outﬂow BC b 0 : ut 0 t £ bux 0 t cu 0 t 0 To discretize (2.2.13) we follow the suggestion in [P3] and generalize it for the θ–scheme: 2

¤ θ b θ

n ¡ n n

£ ¢

(2.2.14a) inﬂow BC b 0 : D u bD u £ c u 0 t 0 x 0 a 0

¤ θ θ

n n n

¦ ¥

£ ¢

(2.2.14b) outﬂow BC b 0 : D u £ bD u cu 0 t 0 x 0 0 ∆ and analogously at the right boundary at L ¢ J x. While Halpern showed that the interior

(Crank–Nicolson) scheme together with the absorbing boundary condition (2.2.14) (for c ¢ 0) is stable and has order two in time and space, the resulting scheme suffers from reduced accuracy as we will see later in the numerical examples of Section 3. Discrete TBC. In order to avoid any numerical reﬂections at the boundary and to ensure unconditional stability of the resulting scheme we will construct discrete TBCs instead of choosing an ad–hoc discretization of the analytic TBC (2.1.8) like (1.2.7) or the approach of Halpern. The discrete TBCs completely avoid any numerical reﬂections at the boundary at no additional computational costs (compared to (2.2.9)). 2.2. Derivation of the DTBC. We mimic the derivation from Section 1 on a discrete level:

we obtain the DTBC by solving the discrete left exterior problem, i.e. (2.2.1) for j 1:

∆t ¤ θ θ θ

∆ n σ∆ ¡ σ ∆ n κ n ∆ n ¦ @ ¥

£ £ ¢

(2.2.15) u £ b 1 u 2 u r u t j ∆x x x j j x j

¤ 2

¦ % ∆ % ∆ κ ∆ σ ¢ where r ¢ a t x denotes the (parabolic) mesh ratio, c t 2 and denotes the param- ∆ ∆

eter explained in (2.2.4). In the scheme (2.2.15) we used the difference operators t ¢ t Dt ,

¤ ¤

∆2 ∆ 2 2 n . ∞ n n ¦ - ¦

∑

¢ ¢ ¢ x D , etc. Once more we apply the –transformation: u uˆ z : u z , x x j j n ) 0 j

¥ j ﬁxed, to solve (2.2.15) explicitly. We assume for the initial data u j ¢ 0 j 2 and obtain the

transformed scheme

¤ ¤ ¤ κ

1 z 1 2 2

¦ τ ∆ τ @ ∆ @ ¦ ∆ ¦ ¥ ¥

£ £ ¢

(2.2.16) uˆ j z £ uˆ j z uˆ j z j 1 x x x θ ¡ θ

r z £ 1 r

with

b∆x ¤ b∆x

@ ¦ τ σ ¥ τ σ ¢ (2.2.17) ¢ 1 a a 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 47 The two linearly independent solutions of the resulting second order difference equation (2.2.16)

take the form

¤ ¤

j 1

¦ ν ¦ ¥ ¥ ¢ (2.2.18) uˆ j z 1 2 z j 1

ν ¤ ¦

where 1 2 z are the solutions of the quadratic equation

¡ ¡

τ τ τ

@ @

2 1 z 1 1 £

¡ ¡

ν ¤ κ ν

£ £ ¢

(2.2.19) 1 £ 0

¡ ¡ ¡ τ θ θ ¦ τ

1 r 2 z £ 1 2 1

¡ ∞ ν

& ¦(&B© Since we are seeking decreasing modes as j we have to require 1 z 1 and obtain

the –transformed left discrete TBC as

¤ ¤

¤ ν

¦ ¦ ¦

(2.2.20a) uˆ1 z ¢ 1 z uˆ0 z

Analogously, the –transformed right discrete TBC reads

¤ ¤

¤ 1 τ

¦ ¦ ¥ ¦ ν (2.2.20b) uˆJ 1 z ¢ 1 z uˆJ z

τ @

1 £

¤ ¤ ¤ ¤

ν ν τ ¡ τ

¦ ¦ ¦ % ¦

@ ¢

where we have used the property 1 z 2 z 1 £ 1 .

REMARK 2.5. It was shown by Lill [D4, Theorem 3.11] that for the solutions to the qua- ¤

ν ¤ ν ¦#&A© & ¦(&A & &A©

dratic equation (2.2.19) & 1 z 1, 2 z 1 holds for z 1.

It only remains to inverse –transform ν1 z ¦ in order to obtain the discrete TBCs from

(2.2.20) and in a tedious calculation this can be performed explicitly:

¤ ¤ ¤ ¤

. ¡

¡ 1

- ¦ ¦ ¦ τ ¦ ν τ ν

CALCULATION (of 1 1 z ). First we reformulate 1 1 2 z appropriately:

¡ ¡ ¤ ¤ τ τ κ

@ 1 z 1

¡ ¡

τ ν ¤

¦ ¦

¢ £ £

1 1 2 z 1

¡ θ θ ¦

2 r r 2 z £ 1

¡ ¡ ¤ τ τ 2 ¤

1 z 1 @

¡ ¡ ¡

¤ κ τ τ

6 ¦ ¦

£ £

1 £ 1 1

θ θ ¦

r 2 z £ 1 2 !

2 ¡ κ

1 1 1 z 1 Az 2Bz £ C

¤ ¤

£ £

¢ H θ 1 ¡

¡ 1

θ ¦ θ θ θ r 2 1 r r 2 z £ 1 z £ θ

∆ ¤ ¦ e %

Here P ¢ b x 2a denotes the cell Peclet´ number. The constants can be determined as ¤

κ 2 θ κ ¤ β 2 ¦ ¦ ¥

¢ £

(2.2.21a) A 4r H £

¡ ¡ ¡ ¡

κ κ @ θ ¦ κ θκ @ β β @(¥

(2.2.21b) B ¢ 2rH 1

¤ ¤ ¤ ¡

2 ¡ 2

@ @

κ @ θ κ β

¦ ¦ ¦ ¥ £ (2.2.21c) C ¢ 4r 1 H with the abbreviations

κ θκ θ ∆ ¥

£ ¢ £

(2.2.22a) ¢ 1 2 1 c t

¤ ¤

¡ ¡ ¡ ¡

¦ ¦ ¥ κ @ θ κ θ ∆ ¢ (2.2.22b) ¢ 1 2 1 1 1 c t

b∆t ¤ b∆t

β θ β @ θ

¥ ¦ ¥ ¢ (2.2.22c) ¢ 1

∆x ∆x

¤ ¡

σ ¦ e £ (2.2.22d) H ¢ 1 1 2 P

σ σ

We note that for a reasonable choice of (”upwind differencing”), i.e. 0 1 2 for b 0

σ σ ¦ and 1 % 2 1 for b 0, 1 2 Pe 0 holds and consequently H 1 and A is always positive. This will be assumed from now on. 48 2. THE CONVECTION–DIFFUSION EQUATION

For the inverse –transformation we use again (1.2.23) and obtain

! ¡

2 ¡ 2

¤ £

Az 2Bz £ C 1 Az 2Bz C

! ¤

¤ λ ¥ ¦

¢ F z µ

¡ ¡

θ θ ¦ θ θ £ 2 z £ 1 A 2z z 1

1 A C 1 E ¤

¤ ¤

! λ ¥ ¦ ¥ £

¢ F z µ

¡ ¡

¦ θ θ ¦ θ θ A 2 2 1 z 2 z £ 1

with E given by ¡

1 θ θ

£ £ (2.2.23) E ¢ A 2B C θ 1 ¡ θ

Now applying the inversion rules (especially (IZ.9)) given in the Appendix yields for 0 1: !

2 ¡

Az 2Bz £ C 1 A C

1 0 1

! ¤

¤ δ δ £

¢ n n ¡

θ ¡ θ θ θ

¦ ¦

2 z £ 1 A 2 2 1 ¡

θ n ¤

E 1 0

¡ ¡

¦ δ 1 £ n Pn µ ¡ θ

2 1 θ ¦ ¤

1 A ¤ C

! ¤

¦ ¦ £

¢ Pn µ Pn 1 µ

¡ A 2θ 2 1 θ ¦

n 1 ¡

θ n k ¤

E 1

¦ ¥

£ ∑ Pk µ ¡

2 1 θ ¦ θ

k ) 0

where 1 denotes the discrete convolution. Finally, we obtain with (IZ.10)

¤ ¡

¡ n ¤

¤ n

κ ¦ θ 1 1 1

1 . 0

¡ ¡

¤ ¤

- ¦

τ ¦ ν δ

¢ £ £ 1 1 2 z H

¡ ¡ ¦

r 2 1 θ ¦ r 2θ 1 θ r θ ¤

1 A ¤ C

6 ¦ ¦

Pn µ £ Pn 1 µ

(2.2.24) ¡ r A 2θ 2 1 θ ¦

n 1 ¡

θ n k ¤

1 1

¦ ¥ £ ∑ Pk µ ¡ 2 2θ 1 θ ¦ θ

k ) 0 since the constant E can be simply determined from (2.2.23) as

1 ¤

(2.2.25) E ¢ ¡

θ 1 θ ¦

We recall from Chapter 1 that the parameters λ, µ are given by !

A B

! !

λ ! ¥ ¢ (2.2.26) ¢ µ

C A C

It can easily seen from (2.2.21) that the following relations hold: ¤

¤ b∆t 2

¡ ¡

¡ 2

∆ θ ¦ ∆ ∆ ¦ ¥

£ £ £ (2.2.27) C ¢ A 4rH 2c t 1 2 4rHc t c t

∆x

¤ ¡ 2 ¡ 2 2 2

¦ e (2.2.28) B AC ¢ 2r H P

For C © 0, i.e. iff ¤ 2 2

1 c ∆x ¦

2 ¡ 2

¤ ¥ e

(2.2.29) H P H £ ¡

2 1 θ ¦ r 2a

2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 49

% ©

we conclude from (2.2.27) that for a rather implicit scheme (θ 1 2) we have λ 1 and from

&3 ©

(2.2.28) we see that in the typical case 2σ & Pe 1 we have µ 1, i.e. the Legendre polynomials

¥ Pn µ ¦ in (2.2.24) have to be evaluated outside the standard domain 1 1 . We note that special routines for that purpose exist [C4].

REMARK 2.6 (“damped” Legendre polynomials). In contrast to the calculation of the con-

0 n volution coefﬁcients in case of the Schrodinger¨ equation in Chapter 1 we have µ © 1 which λ n

makes it essential to use the damping factor to circumvent the “blowing up” of the Legendre

∞ polynomials Pn µ ¦ as n . This behaviour can be seen from the following Lemma:

LEMMA 2.3 (Theorem 8.21.1 (Formula of Laplace–Heine), [S7]). Let µ be an arbitrary real

¡ ∞

¥ or complex number which does not belong to the closed segment 1 1 . Then as n ,

¤ 1 n 2 ¡ 2 ¡ 4

¦ + £ (2.2.30) Pn µ ¢ µ 1 µ µ 1 2πn

This formula holds uniformly in the exterior of an arbitrary closed curve which encloses the

segment 1 1 , in the sense that the ratio tends uniformly to 1. ¤

¤ n

¦ λ The “damped” Legendre polynomials Pn µ : ¢ Pn µ are computed directly by the re-

cursion formula

¤ ¤ ¤

2n £ 1 n

1 ¡ 2

¦ ¦ ¥ ¥ ¦ λ λ

(2.2.31) Pn 1 µ ¢ µ Pn µ Pn 1 µ n 0

n £ 1 n 1

¦ ; with the starting values P0 ; 1, P 1 0. The asymptotic behaviour of the Pn µ will be the

topic of the next subsection. ¤

λ ¤ λ n ¦ ¦ REMARK 2.7 (negative C). If C 0 then , µ become complex, but Pn µ ¢ Pn µ remains a real quantity. This can easily be seen from the recursion formula (2.2.31). Therefore this case needs no special treatment.

REMARK 2.8 (Implicit Euler scheme). Note that the convolution coefﬁcients for the im-

θ θ ¢ plicit Euler scheme ( ¢ 1) are not included as a special case. For 1 the calculation has

to be done separately, since in this case we obtain

1 z 1 1 0 1 1 ¡

¤ δ δ

(2.2.32) ¢ n n

¡ θ θ ¦

2 z £ 1 2 2

The results are given in (2.2.35) below.

To summarize our calculations done so far we write the obtained discrete TBCs:

0 0

¡ n n n n k k

¦ 1 ¥ ¢

(2.2.33a) 1 u ¢ u ∑ u 1 0 0

k ) 1

0 0

n n n n k k

τ @J¦ ¥ ¥ 1

¢ ¢

(2.2.33b) 1 £ uJ 1 uJ ∑ uJ n 1

k ) 1

50 2. THE CONVECTION–DIFFUSION EQUATION

n θ

with convolution coefﬁcients for 0 1 given by !

0 κ

0 1 £ A

£ £ ¢ H

r 2θr

¤ ¡

¡ n

0 n

¦ θ 1 1 1

n 9

¤ !

¢ £

(2.2.34) ¡ 2θ 1 θ ¦ r θ r A

n 1 ¡

¤ ¤

¤ θ n k

A C 1 1

¤ ¤

¦ ¦ ¦ ¥ £

Pn µ £ Pn 1 µ ∑ Pk µ

¡ ¡

2 ¦ 2θ 2 1 θ ¦ 2θ 1 θ θ

k ) 0

for n 1, and in the case ¢ 1: !

0 κ

0 1 £ A

£ £ ¢ H

r 2r

1 1 1 B

¡ ¡

1 ¡

! !

¦ ¥

£ ¢ £ (2.2.35) ¢ AP1 µ 2B 1

2r 2r A 2r A

¤ ¤ ¤

0 1

n ¡

¦ ¦ ¦ ¥ ¥ £

¢ APn µ 2BPn 1 µ CPn 2 µ n 2

2r A ¤

¤ n 0

¦ λ ¦ δ where Pn µ : ¢ Pn µ denotes the “damped” Legendre polynomials and n is the Kronecker symbol.

REMARK 2.9 (Explicit Euler scheme). The calculation above does not work directly in the ¤

θ ν ¦ case of the explicit Euler scheme ( ¢ 0) since the inverse –transformation of 1 z does not exist. In this situation we use instead of (2.2.20) the –transformed discrete TBCs in the

formulation

¤ ¤

1 ¤ 1

¦ ¦ ¦ ¥

(2.2.36a) z uˆ1 z ¢ z 1 z uˆ0 z

¤ ¤

¤ 1 τ

1 1

¦ ¦ ¦ ν (2.2.36b) z uˆJ 1 z ¢ z 1 z uˆJ z

τ @ 1 £

To derive the convolution coefﬁcients of the DTBC we consider

! ¡

¡ 2 ¤ ¤ κ

1 z 1 1 Az 2Bz £ C

¡ τ 1ν 1 ¦ ¦

£ £ £ 1 z 1 z ¢ H z r 2r z 2r z (2.2.37)

¡ 1 2

κ ¤

1 1 A 2Bz £ Cz

¡ 1

¦ ¥

λ ¥

£ £ £ ¢ H z F z µ 2r r 2r 2r A

and calculate ¤ ¤ κ

1 1 0 1 1 ¡

¡ 1

¦ τ ¦ ν δ δ

£ £ 1 z 1 z ¢ H 2r n r 2r n

(2.2.38)

¤ ¤

1 ¤

¦ ¦ ¦ £

APn µ 2BPn 1 µ £ CPn 2 µ 2r A Finally, we obtain the following discrete TBCs for the explicit Euler scheme:

n 1

0 0

0 n ¡ n 1 n k k

¥ τ ¦

(2.2.39a) u ¢ 1 u ∑ u 0 1 0

k ) 1

n 1

0 0

0 n n 1 ¡ n k k

τ @J¦ ¥ ¥ £

(2.2.39b) uJ ¢ 1 uJ 1 ∑ uJ n 1

k ) 1

2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 51

with convolution coefﬁcients n given by

0 1 £ A ¥ ¢ 2r

0 κ 1 B

1 ¡

£ £ (2.2.40) ¢ H 1

r 2r A

¤ ¤ ¤

0 1

n ¡

¦ ¦ ¦ ¥ £

¢ APn µ 2BPn 1 µ CPn 2 µ n 2 2r A

2.3. The Asymptotic Behaviour of the Convolution Coefﬁcients. In this subsection we

0 want to investigate the asymptotic behaviour of the convolution coefﬁcients n given by (2.2.33). As in Chapter 1 we will see that it is beneﬁcial to reformulate the DTBC (2.2.33).

The summed convolution coeffcients. It follows immediately from Lemma 2.3 that limn : ∞

¤ ∞ ¦

Pn µ ¢ 0 holds. Thus the coefﬁcients have the following asymptotic behaviour for n :

¤ ¤

¡ ¡ ¡

¡ n n n 1

0 n n k

¦ ¦ θ θ θ

1 1 1 1 ¡

n ¡

¤ ! ¦ (2.2.41) ¤ ∑ P µ

+ k

¡ ¡

¡ θ θ ¦ θ 2 θ θ 2 1 r 2rθ 1 θ ¦ A 1 k ) 0 Using

n 1

θ k ¤

1 1 1

! ! ¦

lim ∑ Pk µ ¢

¡ : ∞ n A 1 θ A θ 1 θ 2 2 ) λ λ

k 0 ¦

1 £ 2 1 θ µ 1 θ

1 ¢

(2.2.42) ¤

θ θ 2

¦ £

A £ 2 1 θ B 1 θ C

¡ θ ¢ ¢ 1 θ

1 θ E

in (2.2.41) we ﬁnally obtain

¤ ¡

¡ n

0 n

¦ θ n 1 1

¤ θ ∞ ¥

(2.2.43) ¥ 0 1 as n + θ ¡ θ θ

1 ¦ r

θ n ¡ n

% ¦ %

In the special case of the Crank–Nicolson scheme ( ¢ 1 2) we get 4 1 r. This + alternating behaviour (2.2.43) may lead to subtractive cancellation in (2.2.33). Therefore we

prefer to use the following (weighted) summed coefﬁcients in the implementation

0 0 0

n n ¡ n 1 0 0

¥ ¥ ¥ ¥

θ θ ¦ θ θ

£ ¢ (2.2.44) s : ¢ 1 n 1 s : 0 1

and compute with the help of (2.2.23)

¤ ¤

¤ θ θ

1 A 1 A C 1 C

n ¡

¦ ¦ ¦

¢ £ £

s Pn µ £ Pn 1 µ Pn 2 µ

¡ ¡

r A 2 θ 2 1 θ 2 2θ 1 θ ¦ 2

¤ ¤

1 ¤

¦ ¦ ¦ £

(2.2.45) ¢ APn µ 2BPn 1 µ CPn 2 µ

2r A

¤ ¤ A ¤

¡ 1 2

¦ ¦

¦ λ λ £

¢ Pn µ 2µ Pn 1 µ Pn 2 µ 2r 52 2. THE CONVECTION–DIFFUSION EQUATION

Using

¤ ¤

1 ¤ n n 1 2

¦ ¦ ¥

λ ¦ λ £

µ Pn 1 µ ¢ Pn µ Pn 2 µ

¡ ¡ 2n 1 2n 1

we ﬁnally get

¤ ¤

¡ 2

¦ λ

A Pn µ Pn 2 µ

n ¡

¥ ¥

(2.2.46) s ¢ n 2 2r 2n ¡ 1 to use in the DTBC:

n 1

¤ ¤ ¤

¡ ¡ ¡ ¡

¡ n 0 n n k k n 1

¦ ¦ ¥ θ τ ¦ θ θ τ

(2.2.47a) 1 u s u ¢ ∑ s u 1 1 u

1 0 0 1

k ) 1

n 1

¤ ¤ ¤

¡ ¡

n ¡ 0 n n k k n 1

@ @

¦ ¦ ¥

θ τ ¦ θ θ τ

¢ £

(2.2.47b) 1 £ uJ 1 s uJ ∑ s uJ 1 1 uJ 1 n 1

k ) 1 n

REMARK 2.10. We see from (2.2.45) that for n 2 the summed coefﬁcients s coincides with the convolution coefﬁcients (2.2.35) for the implicit Euler scheme, i.e. (2.2.44) also holds θ for ¢ 1.

The recurrence formula for the summed coefﬁcients. In this subsection we shall give

two different derivations for the recursion formula of the convolution coefﬁcients s n . The

ﬁrst one is based on the explicit representation (2.2.46) of s n by ﬁrst calculating a recursion

¤ ¤

¦ ¦

formula for Pn 1 µ Pn 1 µ . The second derivation does not require the explicit form of the

n ν coefﬁcients s but only the growth function 1 z ¦ from the –transformed DTBCs (2.2.20).

FIRST DERIVATION: We consider the recursion formula (1.2.47) adapted for the scaled

¤ ¤

¦ ¦

Legendre polynomials Pn 1 µ Pn 1 µ :

¤ ¤

¡ λ 2 ¦ ¦

Pn 1 µ Pn 1 µ

(2.2.48) ¢

2n £ 1

¤ ¤ ¤ ¤

¡ ¡ ¡ 2 2

λ λ ¦ ¦ ¦ ¦ Pn µ Pn 2 µ n 2 Pn 1 µ Pn 3 µ

1 ¡ 2

µλ λ n 2

¡ £ n £ 1 n 1 2n 3

From (2.2.46) we see the recurrence relation for the summed convolution coefﬁcients:

¡ ¡

2n 1 n 2

n 1 1 n ¡ 2 n 1

λ λ

¥ ¥

(2.2.49) s ¢ µ s s n 2 £

n £ 1 n 1

¥ ¥ which can be used after calculating s , n ¢ 0 1 2 by the formula (2.2.45).

SECOND DERIVATION: Now we present an alternative derivation of the ﬁrst convolution

¥ ¥

coefﬁcients s , n ¢ 0 1 2 and the recurrence relation (2.2.49). The advantage of this alternative ¤ ν approach is that we shall only need the growth function 1 z ¦ from the –transformed DTBCs

(2.2.20). ¤

In this second approach we shall ﬁrst derive a ﬁrst order ODE for the growth function ν1 z ¦ , which is given explicitly in the calculation on page 47. In fact, it is more convenient to consider

∞

¤ ¤ ¤ ¤

¡ n 1 n

¦ ¦ ¦ ν ¦ θ θ τ ν

¢ £

(2.2.50) ˜ z : ¢ z 1 1 z ∑ s z

n ) 1 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 53

Using the constants (2.2.21) and (2.2.22d) it has the explicit form

¤ κ 1 κ 1 1

¡ ¡

ν θ ¡ θ 2

¦ ¦ 6

¢ £ £ £ £ £ (2.2.51) ˜ z z H 1 H Az 2Bz C 1 2 r 2r r 2r 2r

ν ¤ ∞

& ¦(&=©

Here, the sign has to be ﬁxed such that 1 z 1 holds. This can be done for e.g. for z ¢ . dν˜

2 ¡

ν< £ Multiplying ˜ ¢ by Az 2Bz C then yields an inhomogeneous ﬁrst order ODE for

¤ dz

ν˜ z ¦ :

¤ ¤ ¤

A C A

¡ ¡

2 ¡ 1 0

¦ ¦ ¥

ν ¦ ν β β β

¢ ¢ £ (2.2.52) z 2z £ ˜ z z 1 ˜ z z : z B B B with

¤ κ κ

A 1 1

¡ ¡

1 ¡

¦ ¥

β θ θ

£ £ £ £ ¢ 1 H H B r 2r r 2r

(2.2.53)

¤ κ κ

1 C 1 ¡

β 0 ¡ θ θ ¦

£ £ £ £ ¢ 1 H H r 2r B r 2r

ν ¤

¦ % ¢

Its general solution includes ˜ 1 2 z as deﬁned in (2.2.51). Recalling from (2.2.26) that A B

λ % % % λ ¦ ν ν

µ, C B ¢ 1 µ holds and using the Laurent series (2.2.50) of ˜ and ˜ in (2.2.52) immediately yields the desired recursion for the coefﬁcients s n :

¡ 1 0 1

β ¥

s s ¢ µ

1 ¡ 2 1 0 β 0

(2.2.54) ¥

£ ¢ 2 s £ s s

µ λµ

¤ ¤ ¤

µ 1

¡ ¡ ¡

¡ n 1 n n 1

¦ ¦ ¦ ¥ ¥

£ ¢ n £ 1 s 2n 1 s n 2 s 0 n 2 λ λ2 which coincides with (2.2.49). We remark that the starting coefﬁcient of the recursion (cf. (2.2.34)) can be determined

with [S3, Theorem 39.1]:

0 ν κ ¦

0 0 ˜ 1 z 1 £ A

θ θ ¥ θ

¢ ¢ £ £ s ¢ lim H 0 1

z : ∞ z r 2r

STABILITY OF THE RECURRENCE RELATION: For investigating whether the recurrence relation (2.2.49) is well–conditioned we proceed similar to Subsection 2.3 in Chapter 1 and

write (2.2.49) as the second order difference equation

n 1 n n n n 1

¥ ¥

£ ¢ (2.2.55) s £ a s b s 0 n 3

with the coefﬁcients

¡ ¡

2n 1 n 2

n ¡ 1 n 2

λ ¥ λ

¢ ¢

(2.2.56) a ¢ µ b 0

> £

n £ 1 n 1

n n There are two linearly independent solutions s1 , s2 to (2.2.55). To prove that (2.2.49) is well– conditioned we have to show that the seeked solution is not a minimal solution to (2.2.55), i.e.

is a dominant solution.

Since the coefﬁcients a n , b n in (2.2.55) have the ﬁnite limits

B C ¡

n ¡ 1 n 2

λ ¥ λ ¥

¢ ¢ ¢ ¢ ¢ ¢

(2.2.57) a ¢ lim a 2µ 2 b lim b 0

∞ : ∞ n : A n A 54 2. THE CONVECTION–DIFFUSION EQUATION

(2.2.55) is a Poincare´ difference equation with the characteristic polynomial

¤ 2

Φ ¦

¢ £

(2.2.58) t t £ at b

1 2 2 ¡

¦ % © 6 λ

The characteristic polynomial has the two zeros t ¢ µ µ 1 and since µ 1 it is

readily veriﬁed that the zeros have distinct moduli

1 λ 1 2 © © t © t 0 Therefore the classical Theorem of Perron which is an improvement of the Theorem of Poincare´ 1.5 can be applied to distinguish two solutions with distinct asymptotic properties:

THEOREM 2.4 (Perron Theorem, [E2]). If the characteristic polynomial (2.2.58) has zeros

t 1 , t 2 of distinct moduli,

1 2 ¥

t © t

n n

then there exist two linearly independent solutions s1 and s2 of (2.2.55) such that

n 1

sk k

¥ ¥ ¢

(2.2.59) lim ¢ t k 1 2 ∞ n : n

sk n REMARK 2.11 (Minimal Solution). Theorem 2.4 implies that s2 is a minimal solution of

(2.2.55) [E2].

n n 1

From the above we conclude that (2.2.55) has a minimal solution s2 for which limn : ∞ s2

n 2 1 %

s ¢ t , while the limit is t for every other solution. Now it is known that the Legendre ¤

2 ¤ ¦ polynomials Pn µ ¦ and the Legendre functions of the second kind (of order zero) Qn µ satisfy the identical three–term recurrence relation. Thus a second solution to (2.2.55) is simply n

¤ ¤

n j

¦ ¥ ¥ β ¦ (2.2.60) s ¢ Qn µ Qn 2 µ n 2

2 ¡

2n 1 ¤

with some constant β. For Qn µ ¦ we have the following asymptotic behaviour [E2, p. 60] : ¤

¤ π

$ $

$ n 1 2

¡ ¡

1 2 2 ¡ 1 4 2

¦ ¥ ¥ (2.2.61) Q µ ¦ n µ 1 µ µ 1 n ∞ n + 2 for µ not on the real axis between ¡ ∞ and 1, thus in particular for those µ which we are consid-

ering here. From (2.2.61) we obtain immediately

¤ ¤

n ¡

¦ ¦ (2.2.62) λ Q µ Q µ

n n 2 +

1 $ 2 n 2

¡ ¡ 2

π µ µ 1 n

$ n

¡ 1 2 2

¤ ∞

1 n t ¥ n $

2 ¡ 1 4 2

¡ ¡

2 µ 1 ¦ µ µ2 1

n 1 n 2

% ¢

Since one can easily verify that limn : ∞ s2 s2 t it follows directly that the minimal

n n n solution is s2 , and that s1 ¢ s given in (2.2.46) is now a dominant solution, i.e. the problem of determining the required values of the convolution coefﬁcients is well–conditioned [E2]. 2. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 55 2.4. Stability of the resulting Scheme. To prove stability of the numerical scheme with the discrete TBCs (2.2.47) we use the discrete energy method. We concentrate on the case of the

θ σ

% % ¢ Crank–Nicolson scheme ( ¢ 1 2) with central differencing of the convection term ( 1 2), i.e. we consider (2.2.15) in the form

1 1 1

2 n 0 n n ¡ n 2 ¡ 2 2

∆ ∆ e∆ κ ¥ (2.2.63) u ¢ r u rP u 2 u

t j x j x j j

∆0 ∆ ∆

@ £

where x ¢ x x denotes the centered difference operator. We multiply the equation with

§ ? $

n 1 2

¥ ¥ ¥ u j and sum it up for the ﬁnite interior range j ¢ 1 2 J 1, using the summation by parts rule (1.2.70): (2.2.64)

J 1 J 1 1 1 J 1 1 1 J 1 1

¤ ¤ ¤

n n n n n

2 0

¡ ¡

n 1 2 ¡ n 2 2 ∆ 2 2 ∆ 2 2 2 ¦ κ ¦ ¦ e

∑ u j u j ¢ 2r ∑ u j xu j 2rP ∑ u j u j 4 ∑ u j

) ) ) j ) 1 j 1 j 1 j 1

J 1 1 2 1 1 1 1

n n n n n ¡

¡ 2 2 2 2 2

∆ ∆ @ ∆ £

¢ 2r ∑ u 2ru u 2ru u x j J x J 0 x 0

j ) 0

1 1 1 1 J 1 1

n n n n n ¡ ¡ 2 2 2 2 2 2

e e κ ¦

2rPuJ uJ 1 £ 2rPu0 u1 4 ∑ u j

j ) 1

1 1 1 1 1 1

¤ ¤

n n n n n n

¡ ¡ ¡ ¡

¡ 2 2 2 2 2 2

¦ ¦ e e

2ruJ 1 £ P uJ 1 uJ 2ru0 1 P u1 u0 Finally a summation with respect to the time index yields the following estimate for the discrete

2 n 2 J 1 n 2

" & & " ∆ ∑

L –norm (deﬁned by u : ¢ x u ) 2 j ) 1 j

N 1 1 1

∆ ¤

2a t n n n

I ¡

N 1 2 2 ¡ 2 2 2

" I" " ¦ " e u 2 u 2 ∆ ∑ uJ 1 £ P uJ 1 uJ x n ) 0

N 1 1 1

n n n ¡ 2 ¡ 2 2

e ¦ (2.2.65) £ ∑ u0 1 P u1 u0

n ) 0

N 1 1 N 1 1 0

∆ 0

I 2a t n n n n

2 ¡ 2 2 ˜ n 2 2 ˜ n

I" " 1 1 ¥

u ∑ u u £ ∑ u u

2 ∆x J J 0 0 )

n ) 0 n 0

0 0

˜ n n ¡ δ0 ¥ ¢ where : ¢ n, j 0 J, given in (2.2.33). Again, as in the continuous case, it remains to show that the boundary–memory–terms in

(2.2.65) are of positive type. We concentrate on the boundary term at j ¢ J and deﬁne the ﬁnite sequences

1 1

§ §

n n

2 ˜ n 2

1 ¥ ¥ ¥ ¥ ¥ ¥

¢ ¢ (2.2.66) fn ¢ uJ gn uJ n 0 1 N

transformed DTBC (2.2.20) yields

¤ ¤ ¤ ¤

z £ 1

. ¡

ˆ N ¡ ν

¦ ¦ ¦ ¦

- e ¢ fn f z ¢ uˆJ z 1 P 1 z 1

(2.2.67) 2 ¤

1 ¤ ¡

N ¡ 2

¦ 6 ¥

¦ κ

£ £ £ ¢ uˆ z z 1 z 1 Az 2Bz C 2r J 56 2. THE CONVECTION–DIFFUSION EQUATION

N ¤ N n n

¦ & &#©

∑ where uˆ z ¢ u z is analytic on z 0. The zeros z1 2 of the square root above are

J n ) 0 J ¤

1 2 ¡

¦ λ 6 λ

given by z1 2 µ µ 1 with , µ deﬁned in (2.2.26). As remarked on page 48 in the

ˆ ¤ &A© & &A ¦

above in the curly brackets is analytic for & z z1 and continuous for z z1 and therefore f z & © is analytic on & z z1. Note that we have to choose the sign in (2.2.67) such that it matches with

ν ¤

& &

1 z ¦ for z sufﬁciently large. For the second sequence gn we obtain

¤ ¤

z £ 1

. N

- ¦ ¦ ¥ ¢ (2.2.68) g ¢ gˆ z uˆ z

n 2 J

& &T© i.e. gˆ z ¦ is analytic on z 0. Now the basic idea is to use Plancherel’s Theorem Z.4 which gives:

N 2π ¤

1 ˆ ¤ ϕ ¦ ¦

∑ fn gn ¢ f z gˆ z d π iϕ 2 0 z ) e n ) 0

¤ ¤ ¤

(2.2.69) 1 ˆ ¤ ˆ ϕ ¦ ¦ ¦ ¦ £ ¢ f z gˆ z f z¯ gˆ z¯ d π iϕ 2 0 z ) e

π ¤

1 ˆ ¤ ¦ ¥ ¦ ϕ

¢ Re f z gˆ z d iϕ

π 0 z ) e

¤ ¤ ¤

ˆ ¤ ˆ

¦ ¦ ¦ ¦ ¥ ¨ ¢ where we have used the fact that f z¯ ¢ f z , gˆ z¯ gˆ z , since fn gn IR. Using (2.2.67), (2.2.68) we obtain

N π

¤ ¤ 1 ¤ 2

2 . ¡

N ¡

& ¦ ¦ - ¦

& e ν ϕ £ (2.2.70) ∑ f g ¢ z 1 uˆ z 1 P Re 1 z 1 d n n J iϕ 4π 0 z ) e n ) 0

¤ 2

ν ¦ & & £ We remark that the pole of 1 z at z ¢ 1 is “cancelled” by z 1 . From (2.2.70) we conclude

that the discrete L2–norm (2.2.65) is non–increasing in time if

¤ ¤

ϕ . ¡

¡ i

- ¦ ¥ D ¨G ¥ F¥ (2.2.71) 1 Pe ¦ Re ν1 e 1 0 ϕ 0 2π ν iϕ

holds. This property of 1 can be shown in the following way. On the unit circle z ¢ e ,

¤ ¤ ¤

ϕ π ¡ ϕ ¦ % ¦ % ¦ £

0 2 , we have z 1 z 1 ¢ itan 2 and thereby

¤ 1 z 1 1 ϕ

¦ κ κ ¥ ϕ π

£ ¢ £ (2.2.72) y z : ¢ itan 0 2

r z £ 1 r 2 ν ¤

Now 1 z ¦ fulﬁls simply

¤ ¤ ¤ ¤ ¤ ¡

¡ ν 2

¦ ¦U6 ¦ ¦

e¦ e

£ £

(2.2.73) 1 P 1 z 1 ¢ y z y z 2 y z P

and therefore we obtain the requested property

¤ ¤ ¤

¤ κ

. ¡

¡ 2

- ν ¦ ¦ ¦ ¥

e ¦ e

£ £ £ (2.2.74) 1 P Re 1 z 1 ¢ Re y z 2 y z P 0 r

iϕ

ϕ π for z ¢ e , 0 2 . We then have the following main result of this Section:

THEOREM 2.5. The numerical scheme with the DTBCs is stable with the property: J 1

n 1 2 n 1 0 2

" " *" " ¥ (2.2.75) u h : ¢ h ∑ u j u h n 0

j ) 1 3. NUMERICAL RESULTS 57

¤ 3 n

2 ¦

Simpliﬁed DTBC. The rapid decay of the s ¢ O n motivates to restrict (2.2.47) to a convolution over the “recent past” (last M time levels) as an option:

n 1

¤ ¤ ¤

¡ ¡ ¡ ¡

¡ n 0 n n k k n 1

¦ ¦ ¥ θ τ ¦ θ θ τ

(2.2.76a) 1 u s u ¢ ∑ s u 1 1 u

1 0 0 1

k ) n M

n 1

¤ ¤ ¤

¡ ¡

n ¡ 0 n n k k n 1

¦ @J¦ ¥ θ τ @B¦ θ θ τ

¢ £

(2.2.76b) 1 £ uJ 1 s uJ ∑ s uJ 1 1 uJ 1 n 1

k ) n M We note that the stability of the resulting scheme is still not proven yet.

3. Numerical Results In the examples of this Section we want to compare the numerical results from using our new discrete TBC (2.2.47) to the solution using either the discretized TBC of Mayﬁeld [T26] or the approximative absorbing BC of Halpern [P3]. Due to its construction, our DTBC yields exactly

(up to round–off errors) the numerical whole–space solution restricted to the computational

¥ interval 0 L . To measure the induced error (especially at the boundary) we calculate a reference solution

on a much bigger domain (with DTBCs). The difference between the reference solution and

¥ the computed solution is called the reﬂected part. In each example we choose 0 1 as the

computational domain and use the initial data

π V

2 x 0 5 ¡

& &3 ¥ cos for x 0 5 0 45

I 2 0 V 45

¦ ¢

(2.3.1) u x 0 else

¡ ∆ 4

¢ ¢ ¢ Example 1. We used the parameters a ¢ 9, b 100, c 0, the time step t 10

∆ % and the rather coarse space grid x ¢ 1 20 (which makes the difference in the accuracy more clear). Figure 2.1 shows the time evolution of the solution computed with the Crank–Nicolson

σ %

¢ ¢

scheme with DTBCs at x ¢ 0, x 1 and 1 2. Figure 2.2 shows the discrete L –norm ¤

n J 1 n 2 1 $ 2

" & & ¦ " ∆ ∑

u : ¢ x u of the reﬂected part computed with the Crank–Nicolson scheme 2 j ) 1 j

with the new DTBCs. and in Figure 2.3 one can see the results when using the boundary ¢ conditions of Mayfield and Halpern at x ¢ 0, x 1. This illustrates the superiority of our discrete TBCs (2.2.47) (observe the different scales!). The simulation with the discretized TBCs (2.2.9) from Mayfield requires the same numerical effort but the computed solution strongly deviates (especially on coarse grids) from the whole– space solution. The discretization of Mayfield is convergent (if a stability condition is fulfilled). The numerical results with the first order boundary condition (2.2.14) from Halpern are better than the one with (2.2.9) but worse than the results with the simplified DTBC (2.2.76) with

M ¢ 10, which has a comparable computational effort. Example 2. In this example we used again the Crank–Nicolson scheme and central differ-

σ %

encing of the convection term: ¢ 1 2. Now we used a ﬁner grid than before: the time step

5 ¡

∆ ∆ %

¢ ¢ ¢

t ¢ 2 10 and the space grid x 1 160. We reduced the diffusivity: a 0 9, b 100,

¢ c ¢ 0 and computed the solution until T 0 014, i.e. 700 time steps. Figure 2.5 demonstrates once more that using our DTBC we obtain (up to round–off errors) the discrete whole–space solution restricted to the computational interval. Now we observe that the numerical results with the ﬁrst order BC (2.2.14) from Halpern are better than before: we obtain approximately the same error as the simpliﬁed DTBC (2.2.76) 58 2. THE CONVECTION–DIFFUSION EQUATION

Evolution of Solution u 1

0.9 t = 0 t = 0.002 0.8 t = 0.004

0.7 t = 0.006 t = 0.008 0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

FIGURE 2.1. Example 1: Temporal evolution of the solution to (2.1.1) with the

WYX W parameters a W 9, b 100, c 0.

−16 x 10 L^2−Norm of reflected Part 2.5 discrete TBC

1.5

0.5

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.2. Example 1: new DTBC: L2–norm of the reﬂected part (only roundoff–errors occur). 3. NUMERICAL RESULTS 59

L^2−Norm of reflected Part 0.12

discretized TBC L. Halpern BC 0.1 discrete TBC: M=10 discrete TBC: M=20

0.08

0.06

0.04

0.02

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.3. Example 1: Other Approaches: L2–norm of the reﬂected part.

Evolution of Solution u 1

0.9 t = 0 t = 0.002 0.8 t = 0.004

0.7 t = 0.006

t = 0.008 0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

FIGURE 2.4. Example 2: Temporal evolution of the solution to (2.1.1) with

Z WYX W a W 0 9, b 100, c 0. 60 2. THE CONVECTION–DIFFUSION EQUATION

−16 x 10 DTBC: L^2−Norm of reflected Part 2

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.5. Example 2: new DTBC: L2–norm of the reﬂected part (only roundoff–errors occur).

with M W 50 (see Figure 2.6). This is due to the ﬁner discretization and the smaller value of a. Note that the BC of Halpern needs less computational costs than the simpliﬁed DTBC (2.2.76)

with M W 50. Nevertheless, to obtain a smaller error one has to use a (simpliﬁed) DTBC. Example 3. In the third example we used the parameters from Example 2, but now we compute the solution with the fully implicit Euler scheme. Figure 2.8 proves that we obtained the whole–space solution up to round-off errors. In Figure 2.9 one can see that using the BC

of Mayfield induces big numerical reflections. The error lies between the error of using the W simplified DTBC with M W 7 and M 10. Note that Figure 2.10 looks the same as Figure 2.6, i.e. the inner scheme seems to have nearly no influence on the reflected part. Example 4. Here we again used the parameters from Example 2, but now we set the re-

action rate c W 300. Figure 2.11 shows the time evolution of the solution computed with the

Z σ W Z W W θ–scheme θ W 0 8, 0 9 with DTBCs at x 0, x 1. Again Figure 2.12 shows that the computed solution equals the whole–space solution (up to round-off errors). In Figure 2.13 one observes that the numerical results with the discretized TBCs (2.2.9) of Mayfield are comparable to the results with the simplified DTBC (2.2.76) with M between 7 and 10. The results using the first order boundary condition (2.2.14) from Halpern are better: they are comparable to the results with the simplified DTBC with M between 30 and 40. 3. NUMERICAL RESULTS 61

−4 2 x 10 Halpern BC vs. simplified DTBC: L −Norm of reflected Part 2.5

Halpern BC cut = 40 cut = 50 2 cut = 60

1.5

0.5

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.6. Example 2: Halpern BC and simpliﬁed DTBC: L2–norm of the reﬂected part.

−4 x 10 TBC with cut : L^2−Norm of reflected Part 6

cut = 30

5 cut = 40

cut = 50

cut = 60 4 cut = 70

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.7. Example 2: simpliﬁed DTBC: L2–norm of the reﬂected part. 62 2. THE CONVECTION–DIFFUSION EQUATION

−16 2 x 10 DTBC: L −Norm of reflected Part 4.5

3.5

2.5

1.5

0.5

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.8. Example 3: new DTBC: L2–norm of the reﬂected part (only roundoff–errors occur).

−3 2 x 10 Mayfield BC vs. simplified DTBC: L −Norm of reflected Part 8

Mayfield BC 7 cut = 7 cut = 10

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.9. Example 3: Mayfield BC and simplified DTBC: L2–norm of the reflected part. 3. NUMERICAL RESULTS 63

−4 2 x 10 Halpern BC vs. simplified DTBC: L −Norm of reflected Part 2.5

Halpern BC cut = 40 cut = 50 2 cut = 60

1.5

0.5

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.10. Example 3: Halpern BC and simpliﬁed DTBC: L2–norm of the reﬂected part.

Evolution uf Solution u 1

0.9 t = 0 t = 0.001 t = 0.002 0.8 t = 0.003 t = 0.004

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

FIGURE 2.11. Example 4: Temporal evolution of the solution to (2.1.1) with

Z WYX W a W 0 9, b 100, c 300. 64 2. THE CONVECTION–DIFFUSION EQUATION

−16 2 x 10 DTBC: L −Norm of reflected Part 1.4

1.2

0.8

0.6

0.4

0.2

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.12. Example 4: new DTBC: L2–norm of the reﬂected part (only roundoff–errors occur).

−3 2 x 10 Mayfield BC vs. simplified DTBC: L −Norm of reflected Part 3

Mayfield BC cut = 7 cut = 10 2.5

1.5

0.5

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.13. Example 4: Mayfield BC and simplified DTBC: L2–norm of the reflected part. 3. NUMERICAL RESULTS 65

−4 2 x 10 Halpern BC vs. simplified DTBC: L −Norm of reflected Part 2.5

Halpern BC cut = 30 cut = 40 2 cut = 50 cut = 60

1.5

0.5

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 time t

FIGURE 2.14. Example 4: Halpern BC and simpliﬁed DTBC: L2–norm of the reﬂected part. 66 2. THE CONVECTION–DIFFUSION EQUATION CHAPTER 3

The Wide–Angle Equation of Underwater Acoustics

This Chapter is concerned with a ﬁnite difference discretization of so–called wide–angle “parabolic” equations (WAPEs). These models appear as one–way wave propagation approximations to the Helmholtz equation in cylindrical coordinates with azimuthal symmetry and include as a special case the Schrodinger¨ equation. In particular we will discuss the discretization of transparent boundary conditions using the ideas of Chapter 1. In the past two decades “parabolic” equation (PE) models have been widely used for wave propagation problems in various application areas, e.g. seismology [U4], [U5], optics and plasma physics (cf. the references in [U3]). Further applications to wave propagation problems can be found in radio frequency technology [U21]. Here we will be mainly interested in their application to underwater acoustics, where PEs have been introduced by Tappert [U22]. An account on the vast recent literature is given in the survey article [U18] and in the book [U16].

1. Introduction to Underwater Acoustics

\ ]

One standard task in oceanography is to calculate the underwater acoustic pressure p [ z r

\ ] ^

emerging from a time–harmonic point source located in the water at [ zs 0 . Here, r 0 denotes

_ W the radial range variable and 0 _ z zb the depth variable. The water surface is at z 0, and the sea bottom at z W zb. In our numerical tests of discrete transparent boundary conditions (in Section 6) we will only deal with horizontal bottoms (for the treatment of a sloping bottom

see [T21]). However, irregular bottom surfaces and sub–bottom layers can be included by

\ ] simply extending the range of z. We denote the local sound speed by c [ z r , the density by

ρ α [ \ ] [ \ ]a` [ \ ]bW [ \ ]

z r , and the attenuation by z r 0. n z r c0 c c z r is the refractive index, with a

reference sound speed c0 (usually the smallest sound speed in the model). Then the reference W π wave number is k0 2 f c c0, where f denotes the (usually low) frequency of the emitted sound. The situation is shown in Figure 3.1. The pressure satisﬁes the Helmholtz equation

∂ ∂ ∂ ∂ d

1 p d 1 p 2 2

W \ ^ \ (3.1.1) r ρ ρ e k N p 0 r 0 r ∂r ∂r ∂z ∂z 0

with the complex refractive index

[ \ ]W [ \ ] α [ \ ] Z

(3.1.2) N z r n z r i z r c k0

In the far ﬁeld approximation (k0r f 1) the (complex valued) outgoing acoustic ﬁeld

ik0r

ψ e

\ ]W [ \ ] (3.1.3) [ z r k0r p z r e

satisﬁes the one–way wave equation:

X X \ ^ Z ψ W ψ (3.1.4) r ik0 g 1 L 1 r 0

67 68 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS ψ = 0 : reflection 0 r

z r

z ρ s ocean (z,r) : density c(z,r) : sound speed

TBC zb sea bottom : homogenous future : elastic model

FIGURE 3.1. Ocean Model X

Here, g 1 L is a pseudo–differential operator, and L the Schroding¨ er operator

2 1 d

e [ ρ e ] [ \ ] WYX ρ∂ ∂ (3.1.5) L k0 z z V z r

\ ]W X [ \ ] with the complex valued “potential” V [ z r 1 N z r . The evolution equation (3.1.4) is much easier to solve numerically than the elliptic Helmholtz equation (3.1.1). Hence, (3.1.4) forms the basis for all standard linear models in underwater acoustics (normal mode, ray representation, “parabolic” equation) [U1], [U22]. Strictly speak- ing, (3.1.4) is only valid for horizontally stratiﬁed oceans, i.e. for range–independent parameters c, ρ, and α. In practice, however, it is still used in situations with weak range dependence, and backscatter is neglected.

“Parabolic” approximations of (3.1.4) consist of formally approximating the pseudo–diffe- X

rential operator g 1 L by rational functions of L, which yields a PDE that is easier to discretize than the pseudo–differential equation (3.1.4). The name “parabolic” approximation arises from the circumstance that plane wave solutions of (3.1.1) in a homogeneous medium satisfy a dispersion relation, which is quadratic for the linear approximation of the square root operator (cf. [U11]). For a detailed description and motivation of this procedure we re-

λ λ

X X c fer to [U6], [U10], [U11], [U18], [U22], [U23]. The linear approximation of g 1 by 1 2 gives the narrow angle or standard “parabolic” equation (SPE) of Tappert [U22]

ψ ik0 ψ

\ ^ Z (3.1.6) WhX L r 0 r 2 This Schroding¨ er equation is a reasonable description of waves with a propagation direction

within about 15 i of the horizontal. Rational approximations of the form λ

p0 X p1

j [ ]W X λ λ (3.1.7) 1 f λ 1 X q1 1. INTRODUCTION TO UNDERWATER ACOUSTICS 69 with real p0, p1, q1 yield the wide–angle “parabolic” equations (WAPE)

p0 X p1L

X \ ^ Z ψ W ψ (3.1.8) r ik0 1 r 0

1 X q1L In the sequel we will repeatedly require that the slope of the approximating function f should

be negative:

]W X _ Z

(3.1.9) f k [ 0 p0q1 p1 0

W W

Figure 3.2 illustrates that the condition (3.1.9) is quite natural. With the choice p0 1, p1 3 c 4,

Approximation of Square Root Operator 1.2

(1−λ)1/2 SPE: 1 − λ/2 1 WAPE: (1 − 3λ/4)/(1 − λ/4)

0.8

0.6

0.4

0.2

0 −0.2 0 0.2 0.4 0.6 0.8 1 1.2

[ λ ] X λ

FIGURE 3.2. Approximation f of the square root g 1

W X λ g q1 1 c 4 ((1,1)–Pade´ approximant of 1 ) one obtains the WAPE of Claerbout [U4]. In

λ X

[U11] Greene determines these coefﬁcients by minimizing the approximation error of g 1 over suitable λ–intervals. These WAPE models furnish a much better description of the wave

propagation up to angles of about 40 i . To the author’s knowledge, no estimate of the approximation error for such parabolic / wide–angle “parabolic” approximation exists in the literature. Also, higher order analogues of (3.1.7), (3.1.8) [U8], [U13] and split–step Pade´ algorithms [U9] have been successfully used for acoustic problems. While we will restrict ourselves here to the WAPE (3.1.8), we remark that the construction of discrete transparent boundary conditions (see Section 5) could be generalized to higher order PEs and even 3D–problems. In this Chapter we shall focus on boundary conditions (BC) for the WAPE (3.1.8). At the

ψ W \ ]lW water surface one usually employs a Dirichlet (“pressure release”) BC: [ z 0 r 0. At the sea bottom the wave propagation in water has to be coupled to the wave propagation in the sediments of the bottom. The bottom will be modeled as the homogeneous half–space region ρ α z ^ zb with constant parameters cb, b, and b. Throughout most of this Chapter we will use a ﬂuid model for the bottom by assuming that (3.1.8) also holds for z ^ zb, possibly with a different rational approximation (3.1.7) (subject to the coupling condition (3.3.9)). 70 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

REMARK 1.1. Now we want to explain shortly why the rational approximations of the form (3.1.7) are especially appropriate for far–ﬁeld simulations.

ρ ρ 1 m ∞

[ n ] If we assume V , , e L IR then the Schrodinger¨ operator

2 1 d

e [ e ] [ ] WYX ρ∂ ρ ∂ (3.1.10) L k0 z z V z

2 1

W [ n ρ ] with a homogeneous Dirichlet BC at z 0 is self–adjoint in L IR ; e dz with the dense

1 ϕ ρ 1ϕ m 1 ]'o ] [ ]bW [ p [

n n domain D L H0 IR e z H IR . Therefore we have the following spectral

ψI m 2 ρ 1

[ n ] representation for all L IR ; e dz : M ∞

I d

]lW [ ] [ ] [ ] \ ^ \ (3.1.11) ψ [ z ∑ a jϕ j z a λ ϕλ z dλ z 0 V j q 1 b where the ϕ j are the eigenfunctions of L corresponding to the eigenvalues λ j and the ϕλ denote

the generalized eigenfunctions of L to λ. ]

In applications of underwater acoustics the sound speed c [ z is typically larger in the sea

] _ bottom than in the water. Therefore V [ z forms a “potential well” in the water region 0 z _ zb, which typically gives rise to bound states of L that represent the propagating modes of

λ 2 2

_ _ W X _ (3.1.4) and (3.1.8). All of the corresponding eigenvalues satisfy 0 j Vb 1 c0 c cb 1,

σ ∞ W [ ] Wts \ ]

if c0 minz r 0 c z . This is illustrated in Figure 3.3 where c Vb denotes the continuous

λ p W \§Z§Z§Zv\ wyxz[ \ ]

und σp WYu j j 1 M 0 Vb the discrete spectrum.

] \ σ [ ]

V [ z L |

1 σc

σp { Dirichlet { z

BC 0 zb

] σ [ ] FIGURE 3.3. Potential V [ z and spectrum L of the Schrodinger¨ operator

Then the solution at range r ^ 0 for the one–way wave equation (3.1.4) is simply

M ∞

λ ~ λ ~ }

ik0 } 1 j 1 r ik0 1 1 r

\ ]W e e [ ] [ ] e e [ ] Z (3.1.12) ψ [ z r ∑ a j e ϕ j z a λ e ϕλ z dλ V j q 1 b

For λ ^ 1 we have only “evanescent modes”, i.e. exponential damping for the generalized

σ ∞

xs [ ]o[ \ ] xhs \ \ modes in c L 1 . If we assume for a function a that suppa a1 a2 Vb 1 holds then λ

ϕ i z ]W we obtain with λ [ z e for the integral in (3.1.12)

λ ~ λ

}

ik0 1 1 r i z

λ e λ

] W \ [ ] \ (3.1.13) lim a [ e e d 0 pointwise in z ∞ r a1

which follows from the Lemma of Riemann–Lebesgue. From the above we conclude that only X λ

propagating modes contribute in the far ﬁeld both for g 1 and rational approximations. 1. INTRODUCTION TO UNDERWATER ACOUSTICS 71

Therefore a good approximation for λ j is important for the far ﬁeld. Figure 3.4 illustrates the ^ situation showing the Schrodinger¨ modes (λ 1) and the parabolic modes (λ 1).

Approximation of Square Root Operator

Schroed. modes parabol. modes

real imag.

| (1 − λ)1/2 |

1 − λ / 2 σ σ p (4 − 3 λ) / (4 − λ) c

0 0 1 2 3 ↑ V λ b

traveling. modes

] X [ λ λ

FIGURE 3.4. Approximation f of the square root g 1

\ ]

In practical simulations one is only interested in the acoustic ﬁeld ψ [ z r in the water, i.e. for

_ [ \ ] 0 _ z zb. While the physical problem is posed on the unbounded z–interval 0 ∞ , one wishes to restrict the computational domain in the z–direction by introducing an artificial boundary at or below the sea bottom. This artificial BC should of course change the model as little as possible. Until recently, the standard strategy was to introduce rather thick absorbing layers below the sea bottom and then to limit the z–range by again imposing a Dirichlet BC [U6], [U8], [U19], [U20], [U23]. With a carefully designed absorption profile and layer thickness this strategy has been very successful. But without a comparison to the exact half–space solution it is hard to estimate how much an absorbing layer modifies the original problem. Also, absorbing layers increase the computational costs, for SPE– or WAPE–simulations typically by a factor around 2 [U18], [U26]. However, in simulations without attenuation (“false absorbing layer method” [U18], [U26]) or over an elastic sea bottom [U8], much thicker absorbing layers have been used to ensure accuracy and, respectively, numerical stability. In [T18] and [T20] Papadakis derived impedance BCs or transparent boundary conditions (TBC) for the SPE and the WAPE, which completely solves the problem of restricting the z–

domain without changing the physical model: complementing the WAPE (3.1.8) with a TBC

\ ]

at zb allows to recover — on the ﬁnite computational domain [ 0 zb — the exact half–space _ solution on 0 _ z ∞. As the SPE is a Schrodinger¨ equation, similar strategies have been developed independently for quantum mechanical applications [T1], [T6], [T12] (cf. Chapter 1). Towards the end of this introduction we shall now turn to the main motivation. While TBCs fully solve the problem of cutting off the z–domain for the analytical equation, their 72 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS numerical discretization is far from trivial. Indeed, all available discretizations are less accurate than the discretized half–space problem and they render the overall numerical scheme only conditionally stable [T6], [T16], [T19], [T26]. The object of this work is to construct discrete transparent boundary conditions (DTBC) for a Crank–Nicolson ﬁnite difference discretization of the WAPE such that the overall scheme is unconditionally stable and as accurate as the discretized half–space problem. In the literature different WAPEs (or WAPE and the standard “parabolic” equation) have been coupled in the water and the sea bottom. We also analyze in Section 3 under which conditions the resulting hybrid model is conservative for the acoustic energy (weighted L2– norm). While we shall assume a uniform discretization in range r, the interior depth z discretization (i.e. in the water column) may be nonuniform, and we shall discuss in Subsection 5.5 strategies for the ‘best exterior discretization’ (i.e. in the sea bottom). This Chapter is organized as follows: In Section 2 we review the TBCs for the WAPE. In Section 3 we discuss the coupling of the WAPE to the SPE and in Section 4 we derive the TBC for the elastic PE. In Section 5 discrete TBCs are derived and analyzed. Their superiority over existing discretizations is illustrated on several benchmark problems in Section 6.

2. Transparent Boundary Conditions In this Section we ﬁrst discuss the well–posedness of the evolution problem for the WAPE

(in the non–dissipative case α W 0). Then we present the matching conditions at the water– bottom interface which has to be taken into account when we derive afterwards the conservation/decay of the L2–norm. Finally we shortly review the transparent boundary condition for the SPE and sketch the derivation the TBC for WAPE.

2.1. The Well–Posedness of the Evolution Problem for the WAPE. Here we shall discuss the well–posedness of the evolution problem for the WAPE in the critical non–dissipative

case, i.e. for α W 0:

[ ]UX \ ^ \ ^ \ ψ W ψ (3.2.1) r ik0 f L 1 z 0 r 0

ψ \ ]bW subject to the homogeneous Dirichlet BC [ 0 r 0, and with the rational function f given

in (3.1.7). For simplicity of the analysis we only consider the range–independent situation, i.e.

[ ] W [ ] W [ ]

α W α z , ρ ρ z , V V z . ] THEOREM 3.1 (well–posedness of the WAPE). We assume that the refractive index n [ z ,

] ^ [ ] ^ ρ [ ρ the density z 0, and e z are bounded for z 0. Then, the WAPE has a unique solution

2 2 1

] [ ] [ ρ λ n for all initial data in the weighted L –space L IR ; e dz if and only if the pole of f at

˜ 1

W λ W q1 is not an eigenvalue of the operator L with Dirichlet BCs at z 0.

1 m ∞

]

ρ ρ [ n PROOF. In Theorem 3.1 we assumed that V , , e L IR . Then, the Schrodinger¨ operator

2ρ∂ 1∂ d e [ ρ e ] [ ] (3.2.2) L WYX k0 z z V z

2 ρ 1 W [ ] n with a homogeneous Dirichlet BC at z 0 is self–adjoint in L IR ; e dz with the dense domain

1 1 m 1

n n

]lW [ ]o p e [ ] Z [ ϕ ρ ϕ (3.2.3) D L H0 IR z H IR 2. TRANSPARENT BOUNDARY CONDITIONS 73

p0 p1L

e ]lW We now consider the operator f [ L deﬁned as

1 q1L e

∞

λ ]lW [ ] \ (3.2.4) f [ L f dPλ

∞ e with dPλ denoting the projection valued spectral measure of the operator L (cf. [M1], [M7]).

2 1

[ ] [ ρ ] n According to [M1, Theorem XII.2.6] the domain of f L is dense in L IR ; e dz if and only

˜ 1

λ W e [ λ ] [ ] if q1 , the pole of f , is not an eigenvalue of L. In this case f L is self–adjoint and, by

2 1

[ ] [ ρ ] n Stone’s Theorem [M7], ik0 f L generates a unitary C0–group on L IR ; e dz , which yields the unique solution to (3.2.1).

˜ REMARK 2.1. If λ coincides with an eigenvalue λ j of L, then (3.2.1) still admits a unique mild solution for all initial data in the orthogonal complement of ϕ j, the unique eigenfunction corresponding to λ j. Theorem 3.1 generalizes the well–posedness analysis for the WAPE on ﬁnite intervals given in [U2]. There, however, λ˜ can easily lie in the (pure eigenvalue) spectrum of L, what then restricts the class of admissible initial conditions.

1 As q1 is much smaller than 1 in all practical simulations ( 4 in the WAPE of Claerbout; also

λ˜ ∞

\ ] cf. [U11]), usually lies in s Vb , the continuous spectrum of L. Theorem 3.1 then guarantees the unique solvability of the evolution equation (3.2.1) for any initial data. Let us compare the

situation at hand (i.e. the WAPE on the original unbounded interval — and later also the WAPE

s \ with a TBC) to the WAPE restricted to the z–interval 0 zmax with a homogeneous Robin BC at zmax as a simple model for an absorbing layer: there, L has a pure eigenvalue spectrum which inhibits the solvability of (3.2.1) in several cases of practical relevance [U2].

2.2. The Matching Conditions and the Conservation of the Acoustic Power. Now we

turn to the matching conditions and later the TBCs at the water–bottom interface (z W zb). As the density is typically discontinuous there, one requires continuity of the (acoustic) pressure

and the normal particle velocity:

\ ]lW ψ [ \ ] \

(3.2.5a) ψ [ zb r zb r

[ ] [ ] \

ψz zb \ r ψz zb r

e \ (3.2.5b) W

ρw ρb

[ \ ] where ρw W ρ zb r is the water density just above the bottom and ρb denotes the constant density of the bottom.e With these matching conditions (3.2.5) we shall now derive an estimate for the L2–decay of

ρ ρ W [ ] X solutions to the WAPE (3.1.8), z ^ 0. We assume z and apply the operator 1 q1L to (3.1.8):

d 2ρ∂ ρ 1∂ ψ e [ e ] (3.2.6) 1 X q1V q1k0 z z r

d 2 1

X X[ X ] [ X ] e [ e ] Z W ρ∂ ρ ∂ ψ ik0 p0 1 p1 q1 V p1 q1 k0 z z 74 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

ψρ 1 _ _ Multiplying (3.2.6) by ¯ e , integrating by parts on 0 z zb, and taking the real part gives z b 2 1

∂ ψ ρ e p r p dz 0 z

b 2 ρ 1 2ρ 1 ψ W [ X ] s ψ p e X e e s ψ p p 2 p1 q1 k0 Im V dz k Im ¯ 0 w z z q zb 0 e (3.2.7) zb

d 2 1

p p X s s e s ∂ ψ ψ ψ ρ

q1 Re V r 2Im V Im r ¯ dz 0 zb

d 2 2 1 2 1

e e e

e ∂ ψ ρ ρ ψ ψ

p p X s p Z

k dz 2k Re ¯ 0 r z 0 w zr z q zb 0 e ψ ρ 1 Analogously, multiplying (3.2.6) by ¯ r e , and taking the imaginary part we get z

b 2 1

X ] p p e [ ∂ ψ ρ p0 1 r dz 0 zb

1 2 1 d 3 1

e s p p e e e s p

WX ψ ρ ρ ψ ψ 2q1k Im V dz 2q1k Im ¯ 0 r 0 w zr r z q zb 0 e

(3.2.8) zb d

d 2 1

∂ ψ ψ ψ ρ e

[ X ] s p p s s

p1 q1 Re V r 2Im V Im r ¯ dz 0 zb

d 2∂ ψ 2 ρ 1 2ρ 1 ψ ψ e p p e X e e s p Z

k dz 2k Im ¯ 0 r z 0 w z r z q zb 0 e After an easy algebraic manipulation we obtain from (3.2.7), (3.2.8) z

b 2 1

∂ ψ ρ e p (3.2.9) r p dz 0 z 2

b α c0 ˜ 1 1 1 ˜ ˜ ∂ ψ ρ e X e e ∂ ψ ∂ ψ \ WYX ρ 2C1 r dz C1k0 w Im r z r

0 c z q zb e with

[ X ]

2 p1 q1 ˜ d iq1 1

∂ e ∂

\ W Z

C1 W r I k0 r X p1 X p0q1 p1 q1

In the same way a similar equation can be derived for the bottom region z ^ zb: ∞

∂ ψ 2 ρ 1 p e (3.2.10) r p dz zb ∞ 2

c0 ˜ 1 d 1ρ 1 ˜ ˜ WX α ρ e e e ∂ ψ ∂ ψ ∂ ψ Z 2C1 b r dz C1k0 b Im r z r

cb zb z q zb n Adding the two above equations with (3.2.5) gives ∞ 2

2 c0 ∂˜ ψ 1 ] WhX ρ e ∂ ψ [ Z\ α (3.2.11) r r 2C1 r dz 0 c for the weighted L2–norm ∞

2 2 1

] W p [ \ ](p [ ] Z ψ [ Z\ ψ ρ (3.2.12) r z r z dz 0

α ψ α

[SZ\ ] ^ X _ In the dissipation–free case ( 0) r is conserved and for 0 and p0q1 p1 0 it

decays. The discrete analogue of this “energy”–conservation (or –decay for α ^ 0) will be the main ingredient for showing unconditional stability of the ﬁnite difference scheme in Section 5. 2. TRANSPARENT BOUNDARY CONDITIONS 75

W W

REMARK 2.2 (L –estimate for SPE). For p1 1 c 2, q1 0, (3.2.9) simpliﬁes to: z z

∂ b ψ 2 ρ 1 b α c0 ψ 2 ρ 1 1 ψ ψ p p e WhX p p e X s p Z

(3.2.13) dz 2 dz Im ¯

r z q ρ z zb 0 0 c k0 w e 2.3. The Transparent Boundary Condition for the SPE and WAPE. Now we shall review the transparent bottom boundary condition for the SPE and sketch the derivation of the

ψI ψ [ \ ]

TBC for the WAPE. We assume that the initial data W z 0 , which models a point source

\ ] _ _ located at [ zs 0 , is supported in the interior domain 0 z zb (otherwise, if the source is close to the bottom the approach of Section 3 in Chapter 1 can be applied). Also, let the bottom region be homogeneous, i.e. let all physical parameters be constant for z ^ zb. The basic idea

of the derivation is to explicitly solve the equation in the bottom region, which is the exterior

\ ] of the computational domain [ 0 zb . The TBC for the SPE (or Schrodinger¨ equation) (1.1.10) reads now: τ

π r ib

\ X ] 1 ρb ψz [ zb r τ e

ψ i 4 τ \ ]WYX \ (3.2.14) [ z r e d

b π ρ τ g g 2 k0 w 0

W [ X ] with b k0 Nb 1 c 2. This BC is nonlocal in the range variable r and involves a mildly singular convolution kernel. Equivalently, it can be written as τ

π r ib

\ ] ρ ψ [ τ 2k0 i w d zb e e

ψ 4 ibr τ \ ]WYX e \ (3.2.15) z [ zb r π e e ρ d dr r X τ b 0 g 1 and the r.h.s. can be expressed formally as a fractional ( 2 ) derivative [T1], [T6], [T7]: π ρw

ψ 4 i ibr ∂ ψ ibr \ ]lWhX e [ \ ] e Z (3.2.16) z [ zb r 2k0e e r zb r e ρb In [T7] this square root operator is approximated by rational functions which leads to a hierarchy of highly absorbing (but not any more perfectly transparent) BCs for the SPE. By introducing auxiliary boundary variables these BCs can be expressed through local–in–r operators. Hence, this allows for a “local” (2–level in r) discretization scheme [T9]. This scheme, however, introduces numerical reﬂections at the artiﬁcial boundary, whose amplitude depends on the chosen approximation order of the above square root operator.

In order to derive the TBC for the WAPE we consider (3.2.6) in the bottom region: d

d 2 2 1 2

e ] W [ X ] e \ ^ \ [ δ ∂ ψ υ ∂ ψ (3.2.17) b q1k0 z r i b p1 q1 k0 z z zb with

2 2

δ W X [ X ] \ υ W X X[ X ]B[ X ] Z b 1 q1 1 Nb b k0 p0 1 p1 q1 1 Nb After a Laplace transformation of (3.2.17) in r we get

ψ 2 υ δ ψ

[ X ] [ \ ]W [ X ] [ \ ] Z (3.2.18) q1s X i p1 q1 k0 ˆ zz z s k0 i b bs ˆ z s

Since its solution has to decay as z ∞ we obtain υ δ

i b X bs

\ ]'W [ \ ] X [ X ] \ ^ \ ψ [ ψ

(3.2.19) ˆ z s ˆ zb s exp k0 z zb z zb

[ X ] q1s X i p1 q1 k0 and with the matching conditions (3.2.5) this gives ρ υ δ

w i b X bs

\ ]WYX [ \ ] Z ψ [ ψ

(3.2.20) ˆ z zb s k0 ˆ zb s e

e ρ

[ X ] b q1s X i p1 q1 k0 76 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

Here, again denotes the branch of the square root with a nonnegative real part. An inverse

g Laplace transformation [S2] yields the TBC at the bottom for the WAPE :

ρb

\ ]WX η ψ [ \ ] (3.2.21) ψ [ zb r i z zb r ρw

ρ r d

b iθτ iβτ d

\ X ] [ ] [ ] \ βη ψz [ zb r τ e e J0 βτ iJ1 βτ dτ

ρw 0 X

η 1 q1 β p1 X p0q1 k0 θ p1 q1

\ WhX \ W \ W k0 δ δ k0 b 2q1 b q1 where J0, J1 denote the Bessel functions of order 0 and 1, respectively. This is a slight generalization of the TBC derived in [T20] where p0 was equal to 1. Equivalently, (3.2.21) can be written as ρ

1 w ψ \ ]lW η e [ \ ] (3.2.22) ψz [ zb r i zb r ρb ρ r

d 1 w iθτ iβτ

[ \ X ] [ ]UX [ ] Z βη e ψ zb r τ e e J0 βτ iJ1 βτ dτ ρb 0 Both TBCs are nonlocal in r; in range marching algorithms they thus require storing the bottom boundary data of all previous range levels. ∞ We remark that the asymptotic behaviour (for r ) of the convolution kernel in the TBC

3 2

(3.2.15) is O r e , which can be seen after an integration by parts. Using the asymptotic behaviour of the Bessel functions (see (3.5.5)) one ﬁnds that the convolution kernel of (3.2.22)

3 2 also decays like O r e .

3. Coupled Models for Underwater Acoustics

We shall now brieﬂy comment on coupled models for underwater acoustics, as proposed

_ _ ]

in [T19], [T20]. In [T20] the WAPE for the ocean [ 0 z zb is coupled to the SPE for the sea

^ ] bottom [ z zb . In fact, these models are coupled via a TBC corresponding to the SPE, but this is equivalent to the half–space problem. Here we want to point out a mathematical ambiguity of this coupling that may strongly inﬂuence the numerical stability of the discretization scheme. To this end we discuss this coupled WAPE–SPE model in the simple case of constant sound speed c and density ρ, which is rather unrealistic, but it illustrates the situation. 3.1. The coupled WAPE–SPE model. Let us ﬁrst review the WAPE (3.2.1) with the

2 2

e WX ∂ Schrodinger¨ operator L k0 z . When discretizing (3.2.1) one usually applies the operator 1 X q1L to (3.2.1) which gives the following PDE of “Sobolev type” [M6]

ψ ψ

X W X X[ X ] \ ^ \ ^ Z (3.3.1) [ 1 q1L r ik0 p0 1 p1 q1 L z 0 r 0 Since the operators in the numerator and denominator of (3.1.8) commute (even for non– constant c and ρ) this step is mathematically rigorous, and (3.3.1) is easy to discretize (see Section 5). Disregarding for the moment the nonlocality of the involved pseudo–differential operator, one would formally want to write the evolution equation for the coupled model (WAPE and

SPE) as ψ W ψ (3.3.2) r ik0A 3. COUPLED MODELS FOR UNDERWATER ACOUSTICS 77 with

d 2 2

e ∂

p0 p1k0 z

ψ \ _ _ \ (3.3.3a) X 1 0 z z d 2 2 b

e ∂ 1 q1k0 z ψ A W 2

k0e ∂2ψ

^ Z (3.3.3b) \ z z 2 z b

However, the right hand side of (3.3.2) is not well–deﬁned, due to the nonlocality of the pseudo– differential operator in (3.3.3a). Also, its reformulation as in (3.3.1) is not any longer justiﬁed in the coupled case. Even in the dissipation–free case it would result in a non–conservative evolution equation and hence in a non–conservative numerical scheme (nevertheless this strategy is used in [T20]). This is illustrated in Example 3 of Section 6.

Using more involved pseudo–differential operators we will ﬁnd a correct and conservative ] interpretation of (3.3.2), (3.3.3). We ﬁrst consider the pseudo–differential operator f [ L appear-

2 2

e W WYX ∂ ing in the WAPE (3.2.1) with L k0 z . Due to the BC at z 0 it can be expressed in terms of Fourier–sine transforms as ∞ ∞

ψ 2 Φ ξ ψ ξ ξ ξ ] [ ]W [ ] [ ] [ ] [ ] \ (3.3.4) f [ L z y sin y sin z dyd π 0 0 with the symbol

2 2

e X ξ

Φ ξ p0 p1k0 ]W Z (3.3.5) [

2ξ2 e 1 X q1k0 In the coupled WAPE–SPE model one would formally want to write the evolution equation as (3.3.2), (3.3.3). However, as the pseudo–differential operator in (3.3.3a) is nonlocal, acting

n ] _ _ on L [ IR , it cannot be simply restricted to the interval 0 z zb. It is therefore appropriate to deﬁne the coupled evolution equation on the symbol level of the two involved operators (cf. [U10], [M4]). Without attenuation both the SPE and the WAPE conserve the L2–norm and the discrete analogue of this conservation is the main ingredient for showing unconditional stability of the ﬁnite difference scheme in Section 5. Therefore we postulate that the coupled model also has to conserve the L2–norm. This can be achieved if the operator A on the right hand side of (3.3.2) is interpreted as the Weyl operator (see [U10])

2 ∞ ∞ y d z

]W \ [ ] [ ] [ ] (3.3.6) Aψ [ z a ξ ψ y sin ξy sin ξz dydz π 0 0 2

to the symbol

]UX \ _ _ \

Φ [ ξ 1 0 z zb

\ ]W (3.3.7) a [ z ξ 2

k0 ξ2 \ ^ Z X 2 z zb

\ ] As a [ z is real, one readily veriﬁes that the evolution equation (3.3.2), (3.3.6) conserves the L2–norm.

Φ ξ ] Due to the pole of the symbol [ it would be quite difficult to appropriately discretize (3.3.2), (3.3.6), and it is beyond our scope here. We remark that finite difference schemes of pseudo–differential equations with smooth symbols have recently been studied in [F4]. 78 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS However, its discretization using the ideas of Markowich and Poupaud [F5] would be very difficult and will be a topic of future research. 3.2. The coupling of different WAPEs. From the above we conclude that it is not advisable to couple the WAPE and the SPE numerically. As an alternative we shall now analyze couplings of WAPEs with different parameters p0, p1, q1 that can be reformulated as a PDE,

like in (3.3.1). The coupled model

]UX [ ]

p0 [ z p1 z L

X ψ W ψ

(3.3.8) r ik0 1

[ ] 1 X q1 z L is well–deﬁned and can be transformed to (3.3.1) if the numerator and denominator in (3.3.8)

commute. Under the condition

[ ] [ ]W W

(3.3.9) p1 z c q1 z : µ const we can rewrite the pseudo–differential operator in (3.3.8) as

2σ τ [ ]UX [ ] X

]UX [ ]

p0 [ z p1 z L µ z µ z L 1 \

(3.3.10) W 2

[ ] [ ]X [ ] X τ σ 1 q1 z L z µ e z L with

1 2 1

e e

]W X [ ] \ [ ]W [ ] [ ]UX [ ] [ ] Z τ [ σ (3.3.11) z µ p0 z z p1 z p1 z p0 z q1 z Here the numerator and denominator commute, and hence (3.3.8) can be written in the form

2 1

[ n [ σρ ] ] of (3.3.1). The resulting evolution equation is conservative in L IR ; e dz and it allows for a conservative and unconditionally stable discretization (see Section 5 and Example 3 in Section 6).

If the parameters p0, p1, q1 are ﬁxed in one medium, condition (3.3.9) still leaves two free

λ X parameters to choose a different rational approximation model of g 1 for the second medium (cp. [U11]). Hence, one can in fact obtain a better approximation in the second medium than with the originally intended “parabolic approximation”. 4. The Transparent Boundary Condition for an Elastic Bottom In [T3] we presented the following small remark on the coupling of the SPE with an elastic “parabolic” equation (EPE) for the sea bottom [U7], [U12], [U24]. In [T19], [T20] a TBC for this coupling was derived. It reads for the Laplace transformed wave ﬁeld: ρ

b 1 1

\ ]lWhX (3.4.1) ψˆ [ z s

b ρ 4 ] w k0N M [ s

s p

d d

2 2 2

] X [ ] [ ] [ ] [ \ ] \ [ ψ 2Ms s Ns 4 Mp s Ms s Ms s Ns ˆ z zb s with the notation

2 2 2 2

]W X X \ [ ]W X X Z (3.4.2) Mp [ s 1 Np i s Ms s 1 Ns i s

k0 k0 d

d α α W W c Here, Np np i p c k0 and Ns ns i s k0 denote the complex refractive indices for the compressional and shear waves in the bottom. In a tedious calculation this BC can indeed be inverse Laplace transformed (using [S2]) and it reads: r r ωτ ωτ 1

i i 2

\ ]lW [ \ X ] [ ] X [ \ X ] e \ (3.4.3) ψ [ zb r C ψz zb r τ e g τ dτ 2iϕ ψzr zb r τ e τ dτ 0 0 4. THE TRANSPARENT BOUNDARY CONDITION FOR AN ELASTIC BOTTOM 79 with

ρ π b 2 2 k0 k0

4 i 2 2 2

\ ω W X \ ϕ WhX X \ C WYX e Np 1 Np Ns 5 2 ρw 4 π 2 2 k0 Ns

5 3 ϕτ d k0 ϕτ

i 2 2 2 2 i 2

τ τ e τ

]WhX X X X g [ 3 1 e i 3N N 2N e 2 p s s

d d d k 1 1

0 4 2 2 1 4 2 2 2 2

X e W e Z N X N N N N N τ O τ 2 p p s 2 s p s While this inverse transformation was carried out numerically in [T17], [T19], [T20], our analytical TBC may simplify the discretization of this coupled model. However, a discretization of the TBC for a numerical implementation is possible but against the spirit of our approach to derive the BCs on a discrete level. Also the stability of the resulting scheme is unclear. DTBCs for the SPE–EPE coupling (in the spirit of Section 5) are not derived yet, mainly since we do not have a conservative formulation of the coupled elastic–ﬂuid PE. Note that the TBC (3.4.3) for the elastic PE has the same form as the TBC for the SPE

(3.2.14) but the singularity in the integral kernel is stronger (of order 3 c 2).

] W [ \ ] DERIVATION (of (3.4.3)). With the abbreviation ψˆ [ s : ψˆ zb s and the notation

k0 k0 2

[ ]W [ ]WYX s X [ X ] \

(3.4.4a) m s M s i s i N 1 p 2 p 2 p

k0 k0 2

[ ]W [ ]WYX s X [ X ] \

(3.4.4b) m s M s i s i N 1 s 2 s 2 s (3.4.1) reads

4 d 2 2 ] ρ m [ s N 1 k s s 2 2 d

b 0 k0 2 ψ ]WYX X [ ] [ ] [ ] ψ [

ˆ s 4 m s m s N ˆ s

ρ 4 s s s z ] w k0N 2 m [ s k0 k0 s p (3.4.5)

d k0 2 2 ]

ρ [ d

b 8 g 2 ms s 4 Ns k0 2

X [ ] [ ] [ ] \ WYX ψ 4 ms s ms s Ns ˆ z s

ρ 5 2 ] w 4 m [ s 2 k0 Ns p

X ] where we denote the content of the square brackets by f [ s with

σ k0 2

[ X ] Z (3.4.6) W i N 1 2 p We observe that we can write

σ σ d k0 2 2

[ ]WYX s X \ [ ]WX s X [ X ] Z (3.4.7) m s i s m s i s i N N

p s 2 p s

] σ The next step is a shift in the argument of ψˆ z [ s in (3.4.5) by :

d d

b 8 g 2

\ ]WYX [ ] [ \ ] \ (3.4.8) ψˆ [ zb s σ f s ψˆ z zb s σ 5 2 ρw 4 k0 Ns 80 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

W [ ]

X 4

With g i e we obtain for f s

2 d

d k0 2 2 k0 2

[ X ]

X is 2 Np Ns 4 Ns ]lW f [ s π 4 i

e e s (3.4.9) g

d d d k0 k0 k0

4 i 2 2 2 2 2

X [ X ]B[ X ] [ X ]=s \

e s i N N i s i N N s i N 2 p s 2 p s 2 p where

2 d k0 2 k0 2 2

s§? WYXs X s i N i N 2 p 4 s

(3.4.10) 2

d d

d k0 2 2 k0 2 k0 4

WYXs [ X ] s \ s i N N s i N N 2 p s 2 p 16 s

π 2 d 1 k d k0 k0

4 i 0 4 2 2 2

]lW X [ X ] f [ s e N s i N N s i N

s 16 s 2 p s 2 p

d d

d k0 2 2 k0 2

[ X ]s

s i N N s i N 2 p s 2 p

π d k0 k0 1

(3.4.11) 4 i 2 2 2 2

W [ X ]UX X [ X ]

e s i N N g s i N N

2 p s 2 p s s

2 d d k0 2 k0 4 1

s i N N

2 p 16 s s

π 2

d d d 1 k0 k 1

4 i γ γ 2 0 4 W X X \

e s g s s i N N

g s 2 p 16 s s

g g with

γ k0 2 2

[ X ] Z (3.4.12) WX i N N 2 p s Now the transformed transparent boundary condition reads

d d d d k 1 k 1

ψ σ ˜ 0 2 γ γ 0 4 ψ σ ]W X X [ ]

ˆ [ ˆ

s C i N s g s N s

2 p g s 16 s s z

(3.4.13) g

d d

d ˜ 1

γ X γ u ψ [ σ ] w\ X ˆ

C s s s s g

g s z

g with the constant

ρ π b 8 g 2

˜ 4 i Z (3.4.14) C WhX e

5 2 ρw 4 k0 Ns An inverse Laplace transformation of (3.4.13) using (IL.6), (IL.7) yields

r σ σ τ ~

r } r

e e

ψ e ˜ ψ τ τ τ

] W [ X ] [ ] [ r e C z r e g1 d 0

(3.4.15) r ∂

d σ τ ~

˜ } r [ X ] e e [ ] \ C X ψz r τ e g2 τ dτ 0 ∂τ 4. THE TRANSPARENT BOUNDARY CONDITION FOR AN ELASTIC BOTTOM 81 where we compute with the help of (IL.4), (IL.5)

2 d

1 k0 2 d 1 k0 4 1

]We X X

[ τ γ γ g i N s g s N

1 2 p g s 16 s s

g g 2

k0 2 1 d k0 2 1 2 2 1 1

e X X [ X ] e

(3.4.16a) W i N s γ s N N g

2 p g 4 p 2 s s g 2 3 1 k0 γτ d k

2 2 0 2 1 2 2 2

τ e τ

[ X ] [ X ] \ W i N 1 e N N

4 π p 4 π p 2 s

g g

γ d d 1 1 γτ 3 1

1 2 2

e e e

]W X X W [ X ]

[ τ γ γ τ τ g s g s 1 e

2 g s 2 π π

g g g

3 1 γ 1 d 1 γτ d

(3.4.16b) 2 2 2

e e

τ e γτ τ

[ X ] \ W 1 e

2 π 2 π

g g

τ τ q ~ q ~ } g3 } g4

r ∂ σ τ ~

} r

X [ X ] e e [ ] I3 W ψz r τ e g3 τ dτ 0 ∂τ

r τ q r σ τ ~

(3.4.16c) σ τ ~ }

} r r

[ X ] e e k [ ] X [ X ] e e [ ] Z W ψ τ τ τ ψ τ τ z r e g3 d z r e g3

0 τ q 0

~q ~ q ψ } 0 } with z 0 0 Thus the boundary condition (3.4.15) reads now

d σ σ τ ~

r } r

e e k

ψ e ˜ ψ τ τ τ τ ] W [ X ] [ ] [ ] [ r e C z r e g1 g3 d 0

r ∂

d σ τ ~

˜ } r [ X ] e e [ ] C X ψz r τ e g4 τ dτ 0 ∂r

(3.4.17) r

d σ τ ~

˜ } r [ X ] e e [ ] k [ ] W ψ τ τ τ τ C z r e g1 g3 d 0

~ σ τ ~ σ τ }

˜ } r r X ] e e X [ X ] e e [ ] \ C ψzr [ r τ e σψz r τ e g4 τ dτ 0 i.e. r r

˜ στ d στ ]lW [ X ] [ ] [ X ] [ ] \ (3.4.18) ψ [ r C ψz r τ e g˜ τ dτ ψzr r τ e g4 τ dτ 0 0 where

τ τ d τ σ τ ]W [ ] k [ ]X [ ] Z (3.4.19) g˜ [ g1 g3 g4 We calculate 1 3 γτ 5 γτ 3 γ 3

2 2 2

k e e e

τ ]W X [ X ] τ X γ τ X τ g [ 1 e e 3 2 π 2 2 (3.4.20a) g

1 γτ 5 d γτ 3

2 1 2

e e

X [ X ] X [ ] \ W 3 1 e τ 2γ e τ

4 π 2 g

2 σγ 1 1 k 1

2 0 2 2 2 2

σ τ τ e τ

]W W [ X ]B[ X ] \ (3.4.20b) g [ N 1 N N

4 2 π 4 π 2 p p s

g g 82 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS and therefore 3 1 5 1 γτ d 2 γτ

2 2 2 2 1 2 2 2

e e

τ τ e τ τ

]W X X X X g˜ [ ik N 1 e k N N 3 1 e

4 π 0 p 0 p 2 s g

2 d d γτ 3 k 1

2 2 1 2 0 2 2 2 2

e X X X e ik N X N e τ N 1 N N τ 0 p s 2 2 p p s

2 d 1 k d 1

(3.4.21) 0 4 2 2 1 4 2 2 2

X X e W N N N N N N τ

4 π 2 p p s 2 s p s g

3 5 d k0 γτ γτ

2 2 2 2 2

τ e τ

X X X i 3N X N 2N e 3 1 e 2 p s s 1

τ 2 τ [ e ] \ Z W O 0

ϕ ω γ ϕ σ ω W Finally, we deﬁne , by W : i , : i , and set ˜

τ π τ C [ ]W [ ] \ W

(3.4.22) g 4 g g˜ C

4 π g We remark that the well–posedness of the coupled analytic problem is not proven yet and will be a topic of future research.

5. Discrete Transparent Boundary Conditions In this Section we shall discuss how to discretize the TBCs (3.2.14), (3.2.21) in conjunction with a Crank–Nicolson ﬁnite difference scheme for the SPE and the WAPE. Most of the time we shall only consider uniform grids in z and r. While a uniform range discretization is crucial for our construction of discrete TBCs, this construction is independent of the (possibly nonuniform)

z–discretization on the interior domain.

W W W For simplicity we ﬁrst consider the uniform grid z j W jh, rn nk (h ∆z, k ∆r) and the

\ ] approximation ψ ψ [ z r . The discretized WAPE (3.2.6) then reads: j j n

1 n d 2 0 1 0 n

n 2

e e

[ ] X ρ ρ ψ (3.5.1) 1 q1Vj q1k0 jD h j D h D k j 2 2

n d n 1

1 ψ ψ n n d 2 0 1 0 j j

n 2

X X[ X ] [ X ] e [ ρ ] \ W ρ ik0 p0 1 p1 q1 Vj p1 q1 k0 jD h j D h 2 2 2 n 1

n 2

[ \ ] with Vj : W V z j rn 1 and the usual difference operators

n 2 n n

n 1 n ψ X ψ

n ψ ψ X 1 1 j j 0 j j

n n n 2 2

ψ ψ e W \ W Z D k j D h j k 2 h It is well known that this scheme is second order in h and k and unconditionally stable [U2]. Proceeding similarly to the derivation of (1.2.6) one can show 2 ψn 2 j 1 1

1 n n d iq1 1 n 1 n

n 2 ψ 2 ψ ρ WYX e e e \

(3.5.2) D k ∑ C1k0 ∑ Im Vj j k0 Dk j j ρ X

j p1 q1 j ZZ j ZZ

W [ X ] [ X ] with C1 2 p1 q1 c p1 p0q1 . Hence, the scheme (3.5.1) preserves the discrete weighted

2 n 2 n 2

p ψ p ρ

ψ W ∑

c L –norm 2 h j ZZ j j in the dissipation–free case (V real). This also holds when

using a homogeneous Dirichlet BC at j W 0. 5. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 83 5.1. Discretization strategies for the TBC. In the literature three different strategies have been proposed to discretize TBCs, mostly, however, just for the Schrodinger¨ equation. First we review (as in Chapter 1) a known ad–hoc discretization of the analytic TBC.

Discretized TBC of Thomson and Mayﬁeld. In [T26] Thomson and Mayﬁeld used the following discretized TBC for the SPE:

n 1

ψn ψn h ψn e ψn m ψn m ˜ W X k e X e \

(3.5.3) J X J 1 1 J B ∑ J J 1 m e 2 e 2Bk m q 1 with

1 ibmk ] π [ 1 ρb i sin bk e

π 2 4 i 2 bk 2 ˜ ]Ae \ k¡W \ W Z B WhX [ 2 k0 e B e 1 m ρw bk d 1 2 2 m 2 On the fully discrete level this BC is not perfectly transparent any more and it may also yield

an unstable numerical scheme. In analogy to the analytic TBC (3.2.14) it requires the boundary

s \ data from the whole “past range” 0 rn 1 . To compare numerically our discretee TBC for the WAPE to the discretized TBC we use a discretization of the TBC (3.2.21) for the WAPE that is analogous to (3.5.3): r

iθτ iβτ d

\ X ] [ ] [ ] ψz [ zb rn τ e e J0 βτ iJ1 βτ dτ 0

n 1 rm 1 e

θτ d

[ \ X ] [ ] [ ] W ∑ ψz zb rn τ e J0 βτ iJ1 βτ dτ rm m q 0

n 1 n m n m r e

ψ e ψ

X m 1 e J J 1 d θτ

e β β τ [ ] [ ] \

j ∑ J0 rm 1 iJ1 rm 1 e d n n 2 2 h rm m q 0

iz m

] W [ ] with the damped Bessel functions Jν [ z : e Jν z , z C.I This yields the following discretized TBC: n 1

n n ih w ψn e n m n m ˜ ψ X ψ W X k ψ e X ψ e \ (3.5.4) J J 1 J B ∑ J J 1 m

e η ρ b e m q 0 with

] [ θ i θ sin k θ d

2 k 2 ˜ i mk

β \ W [ β ] [ β ] Z

Bk¢W i e 1 m e J0 rm 1 iJ1 rm 1 n θ n 2 2

2 ]

REMARK 5.1. In far ﬁeld simulations one has to evaluate Jν [ z for large complex z, when

numerically calculating these convolution coefﬁcients ˜n. This, however, is a rather delicate ]

problem, and many standard software routines are not able to evaluate Jν [ z for large complex z. p This is due to the exponential growth of the Bessel functions for ﬁxed ν and p z ∞ (see [S1]):

1 2 π π

2 d Im z 1

]W X X £ £ p p \ X _ _ Z (3.5.5) Jν [ z cos z ν e O z π argz π πz 2 4

For this reason we used a subroutine of Amos [C1] (see also [C2], [C3]) to evaluate the damped

] ` β `

Bessel functions Jν [ z , Im z 0 (note that Im 0 for the standard parameter choices in

^ ^ (3.1.7): p1 X p0q1 0 and q1 0). 84 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS In [T16] Mayfield showed for the attenuation–free case that the discretized TBC for the SPE (3.5.3) destroys the unconditional stability of the underlying Crank–Nicolson scheme (cf. Theorem 1.2) and one can expect a similar behaviour for the WAPE. These existing discretizations also induce numerical reflections at the boundary, particularly when using coarse grids. Hence, the existing discretized TBC [T16], [T26] exhibits both stability problems and reduced accuracy, which may require the usage of unnecessarily fine grids.

Semi–Discrete TBC of Schmidt and Deuflhard. In the semi–discrete approach of Schmidt and Deuflhard [T23] a TBC is derived for the semi–discretized (in r) SPE, which also applies for nonuniform r–discretizations and range–dependent coefficients in the exterior domain. This TBC yields an unconditionally stable method (in conjunction with an interior finite element scheme) [T24]. While the semi–discrete approach still exhibits small residual reflections at the artificial boundary, the discrete TBC is reflection–free [T24] (in Subsection 5.5 we shall return to this comparison when discussing the ‘best exterior discretization’). In the recent article [T25] the methods of [T24] are extended to nonuniform r–discretizations and range–dependent poten- tials.

Discrete TBC. Instead of using an ad–hoc discretization of the analytic TBCs like (3.5.3) or (3.5.4) we will construct discrete TBCs for the fully discretized half–space problem, as done in Chapter 1. Our new strategy solves both problems of the discretized TBC at no additional

computational costs. With our DTBC the numerical solution on the computational domain 0 m

j J exactly equals the discrete half–space solution (on j IN0) restricted to the computational domain 0 j J. Therefore, our overall scheme inherits the unconditional stability of the half–space solution that is implied by the discrete L2–estimate (3.5.2).

5.2. Derivation of the DTBC. To derive the DTBC we will now mimic the derivation of

ψ0

\ ` the analytic TBCs from Section 2 on a discrete level. For the initial data we assume j W 0 j

J X 2 and solve the discrete exterior problem in the bottom region, i.e. the Crank–Nicolson ﬁnite X

difference scheme (3.5.1) for j ` J 1 :

d d

d 2 n 1 n 2 n 1 n

n n

X ]lW [ ] \ δ ∆ [ ψ ψ κ ∆ ψ ψ (3.5.6) R b q h j j i R b h j j with 2

2 2k0 h k q1 1

δ W X [ X ] \ W \ W e \

b 1 q1 1 Nb R q k0 X p1 X q1 k 2 p1 q1

k 2

X X[ X ]B[ X ] \ κ W k p 1 p q 1 N b 2 0 0 1 1 b

∆2 ψn ψn ψn d ψn X

where h j W j 1 2 j j 1, and R is proportional to the parabolic mesh ratio. By using

n e the ¤ –transform (for more details see Appendix) : ∞

n n n m

w¥W [ ] W e \ \ p p3^ \ ¤yu ψ ψ ψ (3.5.7) j ˆ j z : ∑ j z z CI z 1

n q 0

(3.5.6) is transformed to

d d

d 2

X ] [ ]WhX [ X ]UX [ ] [ ] Z [ ∆ ψ δ κ ψ (3.5.8) z 1 iq z 1 h ˆ j z iR b z 1 i b z 1 ˆ j z 5. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 85

]¦W [ ] ψ [ ν The solution of the resulting second order difference equation takes the form ˆ j z 1 z ,

X [ ] j ` J 1, where 1 z solves

δ κ d

[ X ]X [ ]

2 iR b z 1 i b z 1 d

X W Z ν X ν d

(3.5.9) 2 1 d 1 0

X ] 2 z 1 iq [ z 1

∞ ν

p [ ](p(_ ¤ For the decreasing mode (as j ) we require 1 z 1. We obtain the –transformed DTBC as

]W e [ ] [ ] \ ψ [ ν ψ

(3.5.10) ˆ J 1 z 1 z ˆ J z e and in a tedious calculation (analogous to the one in Chapter 1, page 13) this can be inverse

transformed explicitly: d

d 1 1 1 e

ν e ν

[ ]3¤ [ ] [ ] [ ] CALCULATION (of 1 iq e 1 z ). First we write 1 iq 1 z appropriately:

(3.5.11)

X ]X [ ]

δ [ κ d

d ν 1 iR b z 1 i b z 1 ] e [ ]W X [ 1 iq z 1 iq 1 d 1 iq

2 e

z 1 iq

d d

X ]X [ ] [ X ]UX [ ] δ [ κ δ κ d

d iR b z 1 i b z 1 iR b z 1 i b z 1

[ ]UX

X 2 1 iq

d d

1 iq 1 iq e 2 e 2

z 1 iq z 1 iq

n n

X ]

[ δ κ δ κ d

d iR b i b z iR b i b iR 1

W X X

d d 1 iq d

2 z eiξ 2 z eiξ 2 z eiξ

d d

d 4 4

δ [ X ]UX κ [ ] δ X [ X ] X κ [ ] z 1 i z 1 q z 1 i z 1

b b b R R b d

δ κ δ d κ 2

[ X ] X

d d

iR b i b z iR b i b iR g Az 2Bz C

W X \

d d 1 iq d 2 z eiξ 2 z eiξ § 2 z eiξ 1 iq

ξ e

with W arg 1 iq and the constants given by

n d

d 4

δ X κ ] δ X κ [ ] \ (3.5.12a) A WI[ i i i 1 iq

b b b b R d

2 d 2 4

δ κ X [ κ δ ] \ (3.5.12b) B W q

b b R b b d

d 4

δ κ ] δ κ X [ X ] Z (3.5.12c) C WI[ i i i 1 iq b b b b R

For the inverse ¤ –transform we use (1.2.23) and the abbreviations (1.2.24) and obtain d

2 d 2

X X

g Az 2Bz C 1 Az 2Bz C λ

W [ \ ] d d F z µ

iξ iξ

[ ]

z e g A z z e

iξ iξ e

1 d e C e E

W X [ λ \ ] \

A d F z µ z iξ g A z e

where the constant E can be determined as:

d d d

d iξ 2iξ iξ 2

e e W X [ ] e Z (3.5.13) E : W A 2e B e C δb iκb δb iκb e 86 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS The inversion rules given in the Appendix now yield

2 d

d d

1 g Az 2Bz C 1 0 iξ 1 n inξ 0

W e [ X ] X [ ] ¤e δ δ δ

d A Ce E 1 e P µ

iξ n n n ¨ n z e g A

n 1

e d

1 d iξ iξ n k

e e

[ ] [ ] X ] [ ] \

W APn µ Ce Pn 1 µ E ∑ e Pk µ

e g A m q 0

where denotes the discrete convolution. Finally, we obtain

d d d d

1 1 d iR 0 iR iξ iξ n

e [ ] W [ ] X X [ ] e [ X ] ¤e ν z 1 iq δ iκ δ δ iκ δ iκ e e 1 2 b b n 2 b b b b

(3.5.14) n 1

e d

iR 1 d iξ iξ n k

[ ] e [ ] X ] e [ ] Z X APn µ Ce Pn 1 µ E ∑ e Pk µ

2 e g A m q 0

˜ 2 2 ˜ 2

W W

Now introducing the notation A W R A, B˜ R B, C R C with the settings

\ WYX \ γ W Rδb σ Rκb we recall from Chapter 1 that the parameters λ, µ are given by

˜ ˜

g A B

\ W \

(3.5.15) λ W µ

˜ ˜ ˜

g g C A g C

with

d d

˜ d ] X [ ] \

d d

[ X ] [ ] \ (3.5.16b) B˜ W γ γ 4q σ σ 4

˜ γ σ γ σ d

~ ~

} n m

n n } n m

ψ ψ ψ e

] W W \ ` \ (3.5.17) [ 1 iq ∑ n 1

J 1 J ¨ J e

m q 1

~ 1 1

} n

]3¤ ν [ ] WI[

with the convolution coefﬁcients 1 iq e 1 z given by

d d i ξ i ξ ~ 0

} n i n in

[ γ X σ ] e δ X [ X ] W 1 iq i e H 1 e 2 n 2

(3.5.18) n 1

e d

d iξ 2 iϕ iξ n m

[ ] e e [ ] e [ X ] e [ ] \

X ζ Pn µ e λ Pn 1 µ ωe ∑ e Pm µ e

m q 0

ϕ 1 X iq i 1

ξ ζ ˜ 2 i 2 ϕ ˜ W \ W p p \ W \

arg d A e argA 1 iq 2

2 d

H d iξ

\ W [ X ] e Z ω W H γ iσ γ iσ e

˜ p p A

0 n

] W [ ] δ [ λ

In (3.5.18) n denotes the Kronecker symbol and Pn µ : e Pn µ the damped Legendre poly-

α W p λ p3W \ sX

nomials (P0 1, P 1 0). In the non–dissipative case ( b 0) we have 1, µ 1 1 ,

[ ](p# ^ p p(^ and hence p Pn µ 1. In the dissipative case αb 0 we have λ 1, µ becomes complex

5. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 87

} n

[ ](p

and p Pn µ typically grows with n. In order to evaluate in a numerically stable fashion it is ] therefore necessary to use the damped polynomials Pn [ µ in (3.5.18). 5.3. The Asymptotic Behaviour of the Convolution Coefﬁcients. Analogously to the considerations in Subsection 2.3 of Chapter 1 one can show that the convolution coefﬁcients (3.5.18) behave asymptotically as

ξ ~

} n n in

X [ X ] \ ∞ \ (3.5.19) W iH 1 e n m m 1

ψ ψ n which may lead to subtractive cancellation in (3.5.17) (note that J j J in a reasonable discretization). Therefore we use the following numerically more stable fashion of the DTBC in the implementation:

n 1

d d

~ ~ } n m

n } 0 n n 1 m

] X WYX [ X ] \ [ ψ ψ ψ ψ

(3.5.20) 1 iq J 1 J 1 iq J 1 ∑ J s

e e

m q 1

~ ~ ~ ~ ~ ξ

} } } }

} n n n i n 1 n

W ` with the summed coefficients s defined by s : e e , n 1. The coefficients s are calculated as

λ 2 [ ]UX [ ]

e d d i P µ P µ ~ 1 n n 2

} n

X [ X ] Z W γ σ δ ζ (3.5.21) s 1 iq i n

2 2n X 1

The recurrence formula for the summed coefﬁcients. In this subsection we shall give ~

two different derivations for the recursion formula of the convolution coefﬁcients s } n . The ~

ﬁrst one is based on the explicit representation (3.5.21) of s } n by ﬁrst calculating a recursion

]X [ ]

formula for Pn 1 [ µ Pn 1 µ . The second derivation does not require the explicit form of the

e ~

} n ν

] ¤ coefﬁcients s but only the growth function 1 [ z from the –transformed DTBCs (3.5.10).

FIRST DERIVATION: We consider the recursion formula (1.2.47) adapted for the scaled

]UX [ ]

Legendre polynomials Pn 1 [ µ Pn 1 µ :

n e

]X [ ] [ λ

Pn 1 µ e Pn 1 µ

e W

(3.5.22) d 2n 1

2 2

]UX [ ] X [ ]UX [ ]

[ λ λ e

1 Pn µ e Pn 2 µ n 2 2 Pn 1 µ Pn 3 µ

e e

λ e λ e X e \ ` Z d µ d n 2

n 1 n 1 2n X 3

From (3.5.21) we see the recurrence relation for the summed convolution coefﬁcients: X

2n X 1 n 2

~ ~ ~

} }

} n 1 1 n 2 n 1

e X e e \ ` \ W λ λ d (3.5.23) s d µ s s n 2

n 1 n 1 ~

} n

\ \ which can be used after calculating s , n W 0 1 2 by the formula (3.5.21).

SECOND DERIVATION: Now we present an alternative derivation of the ﬁrst convolution ~

} n

\ \

coefﬁcients s , n W 0 1 2 and the recurrence relation (3.5.23). The advantage of this alternative

] ¤ approach is that we shall only need the growth function ν1 [ z from the –transformed DTBCs (3.5.10).

ν ] In this second approach we shall ﬁrst derive a ﬁrst order ODE for the growth function 1 [ z . In fact, it is more convenient to consider

∞

d d ~

} n 1 n

] W [ ] X [ ]W e Z (3.5.24) ν˜ [ z : 1 iq z 1 iq ν z ∑ s z

n q 1 e 88 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

It solves the quadratic equation

d d d d d

2 d iR 2

[ ] X X [ X ]X [ ] [ ] X W Z ν˜ X 2 1 iq z 1 iq δ z 1 iκ z 1 ν 1 iq z 1 iq 0 2 b b

and therefore has the explicit form

d d d d d i i i 2

˜ ˜ ]W X [ ] X [ X ]U« X Z ν [ γ σ γ σ ˜

(3.5.25) ˜ ª z z 1 iq i 1 iq i Az 2Bz C

1 2 2 2 2

[ ](p=_ W Here, the sign has to be ﬁxed such that p ν1 z 1 holds. This can be done for e.g. for z ∞. ν

˜ d

d ˜ 2 ˜ X [ ] ν W ˜ ν Multiplying ˜ k by Az Bz C then yields an inhomogeneous ﬁrst order ODE for ˜ z : dz

˜ ˜ ˜ d

A 2 d C A

νk ν β β β

[ ]UX X [ ]W [ ] W \ (3.5.26) z X 2z ˜ z z 1 ˜ z z : 1z 0 B˜ B˜ B˜ e with

d d d

A d i i

X [ X ] X [ ] \ β 1 WX 1 iq γ iσ 1 iq γ iσ e B˜ 2 2 (3.5.27)

d d d

d i C i

X [ γ X σ ] X [ γ σ ] Z β0 W 1 iq i 1 iq i 2 B˜ 2

ν ˜ ] W

[ ˜ c Its general solution includes ˜ 1 ª 2 z as deﬁned in (3.5.25). Recalling from (3.5.18) that A B

˜ [ ]

λ ˜ W λ ν ν

c c

c µ, C B 1 µ holds and using the Laurent series (3.5.24) of ˜ and ˜ in (3.5.26) immedi- ~ ately yields the desired recursion for the coefﬁcients s } n :

~ ~ }

} 1 0 β

X W \ X s s 1 µ e

λ d

d 1

~ ~ ~

} }

} 2 1 0

W \ (3.5.28) X 2 s s s β

µ λµ 0 d

d µ 1

~ ~ ~

} }

} n 1 n n 1

] [ X ] X[ X ] e W \ ` \ X[ n 1 s 2n 1 s n 2 s 0 n 2 λ λ2 which coincides with (3.5.23). We remark that the starting coefﬁcients of the recursion (cf. (3.5.18)) can be determined

with [S3, Theorem 39.1]:

]

ν [ d

˜ z d i i

~ 1 } } 0 0

W W X [ γ σ ]X \ s W lim 1 iq i A ∞ z z 2 2

d d d

d i i i B

~ ~ }

} 1 ν 0 γ σ ζλ 1 γ σ [ ]UX W X [ X ] e W X [ X ] Z s W lim ˜ z s z 1 iq i µ 1 iq i ∞ 1 z

2 2 2 ˜ g A

Simpliﬁed DTBC. Using asymptotic properties of the Legendre polynomials [S7] one ﬁnds

} n 3 2

[ ] ∞ W s O n e , n which agrees with the decay of the convolution kernel in the differential

TBCs (3.2.14), (3.2.21). ~ This decay of the s } n motivates considering also a simplified version of the DTBC (3.5.20). The simplified DTBC can be obtained by cutting off the convolution coefficients beyond an index M, taking only the “recent past” (i.e. M range levels) into account:

n 1

d d

~ ~

} n m

n } 0 n n 1 m e

ψ ψ ψ e ψ

] X WhX[ X ] Z

(3.5.29) [ 1 iq J 1 J 1 iq J 1 ∑ J s

e e

m q n M e 5. DISCRETE TRANSPARENT BOUNDARY CONDITIONS 89 We remark that this numerically cheaper resulting scheme does not conserve the discrete L2– norm (cp. (3.5.2)) (in general, with the DTBC the discrete L2–norm decreases), and hence the numerically stability of the simpliﬁed DTBC is not yet known.

5.4. Discrete Treatment of the Density Jump. So far we did not consider the (typical)

density jump at the sea bottom in the DTBC (3.5.17). In the following we review two possible m discretizations of the water–bottom interface. For the usual grid z j, j IN0 with Jh W zb the ρ discontinuity of is located at the grid point zJ. In this case it is a standard practice [U2], [U20]

to use (3.5.1) with

_ \ ρw \ j J

2ρbρw

\ W \ (3.5.30) ρ W j J j ρb ρw

ρ n ^ Z b \ j J

d 1 n

] [ \ ] W*X [ ] As an alternative one may use an offset grid, i.e. z˜ W¬[ j h, ψ˜ ψ z˜ r , j 1 1 J,

j 2 j j n X where the water–bottom interface with the density jump lies between the grid points j W J 1 and J. For discretizing the matching conditions in this case one wants to ﬁnd suitable approxi-

ψ ρ Ψ ψ ρ ρ ] W [ ] mations for and at the interface z , [ z and z , such that both sides of the

b b e f f b discretized second matching condition (3.2.5b)

n n n n X ψ Ψ Ψ X ψ ψ ψ

1 ˜ J X 1 ˜ J 1 1 ˜ J ˜ J 1

e e Z

(3.5.31) ρ W ρ are equal to ρ c

w h c 2 b h 2 e f f h

] ρ Wz[ ρ ρ This approach results in an effective density e f f w b c 2 (based on a different derivation this was also used in [U7]). In numerical tests we found that the offset grid with the above choice of ρe f f produces slightly better results that have less Gibbs’ oscillations at the discontinuity of ψz at zb. This may be understood by the fact that (3.5.30) requires a higher order derivation (using the evolution equation) than our derivation (3.5.31) (see also [U7], [U19], [U20]). Because of the discontinuity of ψz the higher order derivation yields (slightly) poorer results. Therefore we choose the offset grid for the implementation of the DTBC. At the surface we use instead of

n n ψn WhX ψ W ψ 0 0 the offset BC ˜ 0 ˜ 1. Finally it remains to reformulatee the DTBC (3.5.17) such that the density jump is taken into

account. We rewrite the discretization of the second depth derivative at j W J from (3.5.1): ρ

2 0 1 0 n 2 n d b n n

X X Z ρ ρ ψ W ∆ ψ ψ ψ (3.5.32) h J D h J D h ˜ J h ˜ J 1 ˜ J ˜ J 1 2 2 ρe f f e Comparing the r.h.s. of (3.5.32) to (3.5.6) we observe that only one additional term appears, and

instead of (3.5.8) we get

X ]X κ [ ] ρ δ [

ˆ b z 1 i b z 1 ˆ b ˆ ˆ ]UX X [ ]W [ ]UX [ ] Z

ψ [ ψ ψ ψ d

(3.5.33) ˜ J 1 z 1 iR d ˜ J z ˜ J z ˜ J 1 z

X ] ρ z 1 iq [ z 1 e f f

ψˆ ψˆ ]HW ν [ ] [ ] ν [ ]

Using ˜ J 1 [ z 1 z ˜ J z , where 1 z denotes the solution of (3.5.9), and considering the n

d 1

] [ ] ν [ ν fact that 1 z 1 z is equal to the term in the squared brackets in (3.5.33) we obtain the

¤ –transformed DTBC: ρ ρ

ψˆ ψˆ e f f ψˆ e f f ν 1 ψˆ ]X [ ]W [ ]UX e [ ] [ ] Z (3.5.34) ˜ J [ z ˜ J 1 z ˜ J z 1 z ˜ J z e ρb ρb 90 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS Hence, the DTBC including the density jump reads

ρ ρ

d d d

b b ~

ψn } 0 ψn ] [ ] X X (3.5.35) [ 1 iq ˜ J 1 1 iq 1 ˜ J ρe f f e ρe f f

ρ ρ n 1

d ~

b n 1 b n 1 m } n m

ψ e ψ ψ

\ X ] X[ X ] X WYX[ ˜ ˜ ˜ 1 iq ρ J 1 1 iq 1 ρ J ∑ J s e f f e e f f

m q 1 ~ with the convolution coefﬁcients s } n given by (3.5.21).

5.5. Nonuniform Depth Discretization. At the end of this Section we now address the

question of nonuniform depth discretizations. In the derivation of the DTBC we needed a

` X uniform z–discretization for the exterior problem on z ^ zb, i.e. j J 1. For the interior problem, however, a nonuniform discretization (even adaptive in r) may be used, and this would not change our DTBC (3.5.35). For any given interior z–discretization and a uniform grid spacing hb in the exterior domain, the DTBC will always yield, on the interior domain, the same solution as the corresponding discrete half–space solution. This raises a natural question: given an interior (possibly nonuniform) z–discretization, what is the best uniform discretization of the exterior domain? To analyze this question we ﬁrst consider the three types of errors that are relevant here: Firstly, the error associated with the given

interior discretization does not depend on the choice of hb. In order to avoid strong reﬂections X

due to the nonuniform grid we will assume that the interior grid spacing h j : W z j z j 1 “varies

[ ] [ ] slowly with j” and can be represented as h j W h z j with a “smooth” function h z . To the author’s knowledge, the reﬂections in irregular grids have not yet been theoretically analyzed

for the Schrodinger¨ equation, but very similar effects appear in hyperbolic and parabolic equa- ] tions [F9], [F3]. In numerical tests, however, one can easily verify that discontinuities of h [ z

would introduce spurious numerical reflections of an incident wave (cp. [F9] and references ] therein). Such reflections can be largely reduced by “smoothing” such a discontinuity of h [ z (cf. Example 4 of Section 6). Secondly, the discrete BC at zb may cause outgoing waves to be partially reflected back into the computational domain, and these reflections strongly depend on hb. Finally, for the discretization error of the (uniformly discretized) exterior domain we have to distinguish between traveling waves and evanescent waves. In the first case the discretization error can be interpreted as a modification to the dispersion relation for the outgoing waves (incoming waves are not present in the exterior domain). But the accuracy of their propagation speed is irrelevant, as long as we are only interested in the solution in the interior domain. Hence, the exterior discretization error can be disregarded for outgoing traveling waves. The discretization error of evanescent waves, however, influences the interior solution. Since our DTBC is fully equivalent to a discrete half–space problem, the above discussion of the three error types can be completely reduced to the problem of internal grid reflections for the SPE or the WAPE. In the continuous limit (hb 0) of the exterior discretization, this also holds for the semi–discrete approach of [T23], [T24] for the Schrodinger¨ equation. Following

the above discussion we can now give the best exterior discretization in the ‘traveling wave

[ ] regime’ : the uniform exterior grid spacing hb W h zb generates a completely reflection–free BC and the uniformity of the exterior grid ensures that the outgoing waves will never be reflected back. Their inaccurate resolution in the exterior domain only causes inaccurate wave speeds, but this does not affect the interior solution. 6. NUMERICAL EXAMPLES 91 This behaviour is numerically verified in the simulations of Section 5 in [T24], where a uniformly spaced grid was compared to the semi–discrete approach for the exterior domain. There, a Schrodinger¨ equation with a constant potential is considered, and hence the initial Gaussian wave packet consists only of traveling wave modes in the exterior domain. In the ‘evanescent wave regime’, however, the picture is not that simple, and it is not known yet whether there exists a exists a unique ‘best exterior discretization’. Our simulations of the following Section 6 indicate that it may indeed be advantageous to use a DTBC that corresponds to a finer exterior discretization, as long as the interior and exterior grid spacings are gradually matched to each other.

6. Numerical Examples In the ﬁrst two examples of this Section we shall consider the SPE and the WAPE for comparing the numerical result from using our new discrete TBC to the solution using either the discretized TBC of Thomson and Mayﬁeld [T26] or an absorbing layer. Due to its construction,

our DTBC yields exactly (up to round–off errors) the numerical half–space solution restricted

s \ to the computational interval 0 zb . The simulation with discretized TBCs requires the same numerical effort. However, their solution may (on coarse grids) strongly deviate from the half– space solution. In each example we used the Gaussian beam from [U18] as initial data. Below we present

p p the so–called transmission loss X 10log10 p , where the acoustic pressure p is calculated from (3.1.3).

Example 1. This is a well–known benchmark problem from the literature [U18], [T20],

_ W [T26]. In this example the ocean region (0 _ z 240m) with the uniform density w

3 Z 1 0gcm e is modeled by the SPE (3.1.6). It contains no attenuation and a large density jump

Z ρ W ( b 2 1gcm e ) at the water–bottom interface. Hence, this problem provides a test of the

treatment of the density jump in the TBCs applied along zb W 240m. W The source of f W 100Hz is located at a water depth zs 30m and the receiver depth is at

d 1

W [ ] W p X p e zr 90m. The sound speed proﬁle in water is given by c z 1498 120 z c 60ms , and

1 W the sound speed in the bottom is cb 1505ms e . For our calculations up to a maximum range

1 W

of 20km we used a reference sound speed c0 1500ms e and a uniform computational grid W with depth step ∆z W 2m and range step ∆r 5m (the same step sizes were used in [T26]). In Figure 3.5 the solid line is the solution with our new discrete TBC (3.5.35) and the dotted line is obtained with the discretized TBC (3.5.3). The discretized TBC clearly introduces a systematic phase–shift error, which is roughly proportional to ∆z. The discretized TBC also produces artiﬁcial oscillations (cf. the zoomed region), while our new DTBC yields the smooth solution with the same numerical effort. Figure 3.6 compares the results of our new discrete TBC (solid line) to the solution obtained with an absorbing layer of 240m thickness (dotted line) and a homogeneous Dirichlet BC at

zmax W 480m. Hence the computation took about twice as long as by using the discrete TBC. In our experiments we obtained a better match to the ‘exact’ half–space solution by using the exponential absorption proﬁle

α z X zb [ ]W X λ \ _ _ \ (3.6.1) b z 10 exp 4 1 dB c b zb z zmax

zmax X zb 92 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

Example 1 30

80 Transmission Loss −10*log_10 |p|^2

100 0 2 4 6 8 10 12 14 16 18 20 Range r [km]

FIGURE 3.5. Transmission loss at zr 90m for Example 1: the solution with the new discrete TBC (—) coincides with the half–space solution, while the

solution with the discretized TBC ( ®§®§® ) introduces a phase–shift and artiﬁcial oscillations.

rather than a linear profile. We remark that the profile (3.6.1) and thickness of the absorbing layer were designed as to yield a close match to the ‘correct’ solution. Without such an ‘a– posteriori data fitting’, however, a calculation with an absorbing layer would usually yield a solution with a somewhat larger deviation than suggested by Figure 3.6. With a thicker layer one can of course still improve the results of Figure 3.6, e.g. no more artificial oscillations are visible when using a layer of 760m. Figure 3.7 shows the significant deviations of the solutions using either the discretized TBC or an absorbing layer of 240m from the computed half–space solution, which coincides with the solution using our new DTBC. Example 2. This example appeared as the NORDA test case 3B in the PE Workshop I [U15], [U18], [T20], [T26]. The environment for this example consists of an isovelocity water

1 1

¯ ° ± ± column (c z 1500ms ) over an isovelocity half–space bottom (cb 1590ms ). The den-

3 3

ρ ² ρ

± ±

sity changes at zb 100m from w 1 0gcm in the water to b 1 2gcm in the bottom.

² The source and the receiver are located at the same depth near the bottom: zs zr 99 5m.

The source frequency is f 250Hz. The attenuation in the water is zero, and the bottom at-

³ ³ α ² λ λ tenuation is b 0 5dB b, where b cb f denotes the wavelength of sound in the bottom. 6. NUMERICAL EXAMPLES 93

Example 1 30

80 Transmission Loss −10*log_10 |p|^2

100 0 2 4 6 8 10 12 14 16 18 20 Range r [km]

FIGURE 3.6. Transmission loss at zr 90m for Example 1: the solution with

an absorbing layer of 240m ( ®?®§® ) is quite satisfactory in comparison to the ‘exact’ solution computed with the discrete TBC (—). It is in phase but shows some artiﬁcial oscillations and overestimates the transmission loss at 6km, 7km, and in the range 16–19km.

Example 1 0.04

new discrete TBC discretized TBC absorbing layer 0.03

0.02 |psi|

0.01

0 0 60 120 180 240 depth [m]

´ ¯ µ °(´ FIGURE 3.7. Vertical cut of the 3 solutions at r 19km for Example 1: z r 19km

Here, the steepest angle of propagation (which is the equivalent ray–angle of the highest of the

11 propagating modes) is approximately 20 ¶ (cf. [U15], [T26]). Since the source is located near 94 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS the bottom, the higher modes are signiﬁcantly excited. Therefore the wide–angle capability is important here and we use the WAPE (3.1.8) (with the coefﬁcients of Claerbout) to solve this benchmark problem.

The maximum range of interest is 10km and the reference sound speed is chosen as c0

∆ ² ±

1500ms . The calculations were carried out using the depth step z 0 25m and the range ∆ ² step r 2 5m. Since the source is placed close to the bottom, the TBC was applied 10m below the ocean–bottom interface (the same was done in [T26]).

Example 2

90 Transmission Loss −10*log_10 |p|^2

new discrete TBC 100 discretized TBC

5 6 7 8 9 10

Range r [km] ²

FIGURE 3.8. Transmission loss at zr 99 5m for Example 2: the solution with the new discrete TBC coincides with the half–space solution, while the solution with the discretized TBC still deviates signiﬁcantly from it for the chosen discretization.

The typical feature of this problem is the large destructive interference null at a range of 7km. Figure 3.8 compares the transmission loss results for the discrete and discretized TBCs in the range from 5 to 10km. In a second comparison we extended the computational domain up to 200m. With the given bottom attenuation this 100m layer is thick enough to yield the reasonable approximation shown in Figure 3.9. Figure 3.10 shows the deviation of the solutions with the discretized TBC and with the absorbing layer from the computed half–space solution, which coincides with the solution using our new discrete TBC. 6. NUMERICAL EXAMPLES 95

Example 2

90 Transmission Loss −10*log_10 |p|^2

new discrete TBC 100 absorbing layer

5 6 7 8 9 10

Range r [km] ²

FIGURE 3.9. Transmission loss at zr 99 5m for Example 2: in comparison to the exact half–space solution, the truncation of the computational domain at 200m (the given bottom attenuation then represents an absorbing layer of 100m) introduces a slight phase shift.

Example 2 0.16

new discrete TBC discretized TBC absorbing layer 0.12

0.08 |psi|

0.04

0 0 20 40 60 80 100 depth [m]

ψ ´ ¯ µ °#´ FIGURE 3.10. Vertical cut of the 3 solutions at r 7km for Example 2: z r 7km Example 3. In this example we illustrate the theoretical ﬁndings of Section 3 on coupled models. We use the physical parameters of the ﬁrst two examples but different models for the water and the bottom region. 96 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS We start with considering the environment of Example 2 and compare the results of different

3 1

model couplings. First we ﬁx the WAPE of Claerbout (CWAPE; p0 1, p1 4, q1 4) in the bottom and choose a different (and in fact better) rational approximation (GWAPE) for the water

region that fulﬁls the coupling condition (3.3.9): p1 3q1. The two remaining parameters p0, q1

· °

are then determined by minimizing the approximation error of ¯ 1 λ 2 (in the maximum norm)

¸ ² µ ² ¹ ²

over the interval 0 0008 0 103 , which contains the discrete spectrum of L: p0 1 0000071, ²

q1 0 2501753. We compare this approximation to the case of also using the CWAPE in the water. Furthermore, we show the results when using the SPE in the sea bottom (which clearly violates (3.3.9)) and when using the SPE in both regions.

Example 3

90 GWAPE / CWAPE

Transmission Loss −10*log_10 |p|^2 CWAPE / CWAPE CWAPE / SPE 100 SPE / SPE

7 8 9

Range r [km] ²

FIGURE 3.11. Transmission loss at zr 99 5m in several coupled models (water and sea bottom) for the simulation of Example 2.

Figure 3.11 displays a comparison of the transmission loss from 6.5 to 9km for these different couplings. It turns out that the solution for the coupled GWAPE/CWAPE model is very close to the one using the CWAPE in both media. While the CWAPE/SPE model violates the coupling condition, it only deviates from the above solutions by a slight phase–shift that is typical for the SPE in this example (cp. also the “pure” SPE model).

Now we turn to the dissipation–free situation of Example 1 and focus our attention on a

¯ ° ³ ¯ °

conservative discretization of coupled models that satisfy the coupling condition p1 z q1 z

2 1

º ¯ ° ° ¯ σρ ±

µ const, and hence preserve the L IR ; dz –norm (see Section 3). As a discrete 6. NUMERICAL EXAMPLES 97 analogue of (3.2.9) we obtain in the dissipation–free case

2 J 1 ψn 1

± ˜ j 2 n 1

º 2 n

· ¯ · ° » ψ ψ »

(3.6.2) hD ∑ Im p1 q1 ˜ iq1k D ˜ ¾

k 2 J 1 ½ 0 k J 1 ± σρ˜ j p k ρ ± j ¼ 0 1 0 e f f n 1

º 2 1 n

· ° » » » µ

¯ p1 q1 D ψ˜ iq1k D D ψ˜ ½

¾ h J 1 0 k h J 1

± ±

ρ ρ σ 2

¯ ° ³¿¯ · ° with ˜ j z˜j and p1 p1 p0q1 . Analogously, a discrete version of (3.2.10) can be

shown for the bottom region j À J:

∞ n 2 ψ˜ 1 j 2 n 1 b b º 2 b n

ψ ± ¯ · ° » ψ »

(3.6.3) hD ∑ Im p q ˜ iq k D ˜ ¾

k b 2 1 1 J 1 ½ 1 0 k J 1

σ ρ ± ° b b ¯ p k ρ j ¼ J 1 0 e f f n 1

b b º 2 b 1 n

ψ ± ψ · ° » » » µ

¯ p q D ˜ iq k D D ˜ ½

¾ 1 1 h J 1 1 0 k h J 1

± ±

b 2 b b b

° ³¿¯ · ° σ ¯ σ with b p1 p1 p0 q1 . For coupled models usually takes different values in the water and bottom regions. It follows from (3.6.2), (3.6.3) that the weighted discrete L2–norm on

j Á IN0 is preserved:

J 1 ψn 2 ∞ ψn 2

Â Â

± ˜ ˜

ψn 2 j j (3.6.4) ˜ h ∑ h ∑ const

σρ˜ j ½ σbρb ¼ j ¼ 0 j J provided that the coupling condition (3.3.9) is fulﬁlled. Figure 3.12 illustrates that the discrete L2–norm (3.6.4) is conserved as long as the coupling condition (3.3.9) is satisﬁed. In all four simulations we used the WAPE of Claerbout for the

water region and different models in the sea bottom: only the hybrid WAPE–model with con-

¯ ° ³ ¯ °

stant p1 z q1 z µ 3 renders the scheme conservative (only for this numerical illustration

² ² we choose the values p0 0 6, q1 0 2). A coupling to the SPE (like in [T20]) or to a WAPE in

b b

the bottom with p1 q1 µb Ã 3 all yields a non–conservative scheme. We point out that these schemes are not only non–conservative for the particular norm (3.6.4) but also for any other weighted L2–norm. In the simulations for Figure 3.12, the second sum of (3.6.4) (for the exterior of the computational domain) was evaluated via (3.6.3). Example 4. In this example we illustrate our discussion from Subsection 5.5 on the ‘best uniform exterior discretization’ for the case of evanescent waves. We want to answer the following question: given a uniform interior discretization, can the result from a globally uniform z–discretization be improved by choosing a ﬁner exterior z–discretization, or, equivalently, by

using a DTBC that corresponds to such a ﬁner discretization?

µ Ä As a model problem for this test we consider the SPE (3.1.6) on z Ä 0 r 0 with a homoge-

¯ ° µ Å Å

neous Dirichlet BC at z 0, k0 2m and the “potential well” V z 0 0 z zb 100m,

¯ ° ² µ Ä Å ² ±

ÇÆ

V z Vb 0 3 z zb. In this example, plane waves with a wave number k kcrit 1 2m Ä are evanescent in the exterior domain z Ä zb, and k kcrit transmits a traveling wave into the

2 2 1

¯ · ² ¯ · ° °

± ±

exterior. We choose here the Gaussian beam exp ikz 0 003m z 50m with k 1m as an initial condition. For this choice of k ‘most’ of the Fourier components of this wave cor- respond to evanescent modes in the bottom. Hence, this wave will be predominantly reﬂected back into the interior domain. 98 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

Example 3 1.4

1.35

1.3 WAPE: p_1/q_1=3 SPE 1.25 WAPE: p_1/q_1=3.25 WAPE: p_1/q_1=2.75 1.2

1.15

1.1 (normalized) weighted L2−norm 1.05

0.95 0 2 4 6 8 10 12 14 16 18 20 Range r [km]

FIGURE 3.12. Coupled WAPE–models conserve the discrete L2–norm (3.6.4)

³ only when satisfying the coupling condition p1 q1 µ const (—).

Figure 3.13 compares the effect of choosing different (uniform and nonuniform) z–discreti-

zations. We show the results of this simulation at the range r 200m when the wave packet

has been reﬂected back from the water–bottom interface. The solid line was obtained with the ²

uniform z–discretization h0 0 05m, and it will serve as our ‘exact’ reference solution. The ²

dashed line shows the solution with the uniform grid spacing h1 0 25m. In the following comparisons we will keep this coarser interior grid and will vary the uniform exterior grid.

Following our discussion from Subsection 5.5, we used a gradual transition between these two

¯ ° grid spacings in the depth interval 100 · 110m (piecewise linear grid spacing function h z ).

The dotted curve of Figure 3.13 gives the results with the ﬁner exterior z–discretization ²

h2 0 1m. Close to the sea bottom it shows signiﬁcant improvements over the uniform dis- Å cretization with h1. In the interval 0 Å z 60m both curves almost coincide as the interior discretization error is dominant there, and it implies inaccurate wave speeds that are reﬂected in

the clearly visible phase shift. The dotted curve still exhibits this phase shift up to the sea bot- Å tom at 100m, but for the dashed curve the error in the interval 80m Å z 100m is dominated by the effect of the exterior discretization. It thus seems that the effect of the reduced exterior discretization error (due to the ﬁner exterior discretization) may outweigh (in the interior domain!) the additional reﬂection errors incurred by the nonuniform grid.

µ °

The L ¯ 0 100 –errors (w.r.t. the solid curve) of the solutions with the uniform h1–discretiza-

² ² tion and the nonuniform h1 ³ h2–discretization are, respectively, 0 0370 and 0 0267. Using an even ﬁner exterior discretization does not seem to improve the result much further (L2–error 6. NUMERICAL EXAMPLES 99

Example 4 1 uniform dz=0.05m uniform dz=0.25m nonuniform dz=0.25−0.1m 0.8

0.6 |psi|

0.4

0.2

0 0 20 40 60 80 100 depth [m]

FIGURE 3.13. The ‘best uniform exterior z–discretization’ may be ﬁner than

the interior discretization. Vertical cut of the three solutions at r 200m for

Example 4: the solution ( ®§®?® ) calculated on a nonuniform grid (ﬁner grid in the exterior domain than in the interior) is more accurate in the interior domain than the solution obtained on a uniformly coarse grid (– – –). The reference solution

(—) was calculated on a uniformly fine grid. ³ 0 ² 0266 for the h1 h0–discretization). A finer exterior discretization would, however, require a thicker region to adapt the two grids. We thus conclude that finer exterior discretizations may indeed be advantageous in the case of evanescent waves, and for large ranges these are the important modes in the considered applications of underwater acoustics. 100 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS Conclusions and Perspectives

We have derived a new discretization (discrete TBC) of the analytic TBC for the convection– diffusion equation and Schrodinger¨ –type pseudo–differential evolution equations in one space dimension. It is of discrete convolution form in t involving the boundary data from all the past time levels.

n É The convolution coefﬁcients s È can be calculated via a simple three–term recurrence rela-

3 Ê 2

¯ ° tion and they decay like O n ± . Since our new DTBC has the same convolution structure as existing discretizations, it requires the same computational effort but improves two shortcom- ings: DTBCs are more accurate (in fact, as accurate as the discrete whole–space problem) and they retain the stability of the underlying ﬁnite difference scheme. The underwater acoustics community took a great interest in the results of Chapter 3. Espe- cially the (continuous) TBC for an elastic bottom caught their attention and will be discretized and implemented in existing numerical codes. Our comments on the mathematical difﬁculty of coupling different models for the water and bottom region are also of interest. In underwater acoustics the computations must be very fast and therefore coarse grids are preferred which makes the usage of DTBCs advantageous. We point out that the superiority of DTBCs over other discretizations of TBCs is not restricted to the presented special types of partial differential equations or to our particular interior discretization scheme (see e.g. [T1], [T2], [T3], [T4], [T5], [T10], [P2], [T11]). The

crucial point of our derivation was to find the inverse Ë –transformation of the transformed DTBC explicitly. In more general applications, e.g. 2D transient Schrodinger¨ equation (corresponds to 3D–problems in underwater acoustics), systems of equations or high–order schemes, it might be necessary to use our second approach of deriving the recurrence relation for the convolution coefficients or to calculate the convolution coefficients through a numerical inverse

Ë –transformation (see Section 4 in Chapter 1), but this does not change the efficiency of the presented method. Moreover, in the general case the DTBC will be nonlocal in more than one variable and one has to seek for approximate local BCs that lead to a stable scheme. As a general philosophy, DTBCs should be used whenever highly accurate solutions are important. Our strategies have already been adapted by a Princeton workgroup to systems of wave equations for materials with cracks. Topics of future research include DTBCs for higher order schemes, e.g. the Numerov scheme [U11], or more efficient algorithms like split–step Pade´ schemes [U9] in the application to underwater acoustics. Moreover we will use the ideas of Section 3 in Chapter 1 and derive a DTBC for the WAPE in the case that the starting field is not supported inside the water region (which is the computational domain). We intend to apply the ideas of Markowich and

Poupaud [F5] in order to derive a ﬁnite difference discretization directly for the square root ·

operator Æ 1 L from Chapter 3. Furthermore we search for a conservative formulation of the elastic/ﬂuid coupling which is needed for stable discretizations and as guidance for deriving a DTBC for an elastic bottom.

101 102 CONCLUSIONS AND PERSPECTIVES We will construct DTBCs for situations in which the exterior problems have variable coefficients but still are explicitly solvable. For a linear Crank–Nicolson–type scheme to solve the radial Schrodinger¨ –Poisson system preserving both the energy and the mass [F6] we shall try ∞ to design an appropriate boundary condition at the origin and a DTBC for r Ì [F7]. Also we will design DTBCs for the Schrodinger¨ equation with a linear potential term in the exterior domain. This derivation is based on the knowledge of the asymptotic behaviour of solutions to a discrete Airy equation (cf. [E4], [E5]). This problem arises in underwater acoustics in [T8] and in “parabolic equation” predictions for radar propagation in the troposphere [T13]. Finally to remedy the deficiency that our DTBC is nonlocal in the time variable which is noticeable especially in long–time calculations we construct new approximative transparent boundary conditions [D1]. These BCs are an efficient convolution by an exponential approximation: only one simple update is needed in each time step to compute the discrete convolution. We also prove criteria for the numerical stability of the resulting IBVP. Appendix

The Laplace Transformation

ˆ ˆ

¯ ° ¯ ° Here we discuss the Laplace transformation f f s of a function f f t and present some elementary results. In this thesis the Laplace transformation was used to explicitly solve

the exterior problems in order to derive the continuous TBCs. Ì DEFINITION L.1 (Laplace transformation [S3]). If for a function f : IR» CI there exists a

σ Á IR, such that the integral ∞ ÍÏÎ st

ˆ ±

Ð¿¯ ° ¯ ° ¯ ° µ (L.1) f s : f s : f t e dt 0

ˆ Ä ¯ ° exists for all s Á CI with Re s σ, then f s is the Laplace transformation of f . The most important property of the Laplace transformation is the relationship between de-

rivative and transformation: Ì THEOREM L.1 (Differentiation Theorem [S3]). If the function f : IR» CI is differentiable

and the Laplace transformation of f Ñ exists, then the limit

¯ ° ¯ ° µ

f 0 : lim f t ½

t Ò 0

º σ Ñ

exists and for Re s Ä the Laplace transformation of f fulﬁls ÍÏÎ

Ñ ˆ ° ¯ °U· ¯ ° ² Ð¿¯

(L.2) f s s f s f 0 ½ The initial values enter into the transformation of the derivative.

The essential rule for deriving the continuous TBCs is the Laplace transformation of a con-

Ó µ Ì volution f g of two functions f g : IR» CI which is deﬁned as

¯ Ó °B¯ ° ¯ τ ° ¯ · τ ° τ ²

f g t f g t d 0

THEOREM L.2 (Convolution Theorem [S3]). If the Laplace transformation (L.1) exists for

µ Ì the functions f g : IR» CI and at least one Laplace integral is absolutely convergent then we have

ÍÏÎ ˆ

Ó Ð¿¯ ° ¯ ° ¯ ° ²

(L.3) f g s f s gˆ s

It is also necessary to have a link between norms in the physical and transformed space: Ì THEOREM L.3 (Plancherel’s Theorem [M5]). If the function f : IR» CI is continuous and satisﬁes an estimate

2 Ô σt

¯ °(´ µ À µ ´ f t C e t 0

103 104 APPENDIX for some real constants C, σ, then the Laplace transformation of f (L.1) is an analytic function

for Re s Ä σ and ∞ ∞ 2ηt 2 1 2

± ˆ ¯ °(´ ¯ ° µ Ä ² ´ η ξ ξ η σ

(L.4) e f t dt f i d

0 2π ∞ ½ ± holds.

The Inverse Laplace Transformation Here we shortly give the basic facts about the inverse Laplace transformation which are needed for stating the continuous TBC in physical variables. The inverse Laplace transformation is given as follows:

THEOREM L.4 (Inverse Laplace transformation [S3]). The inverse of the Laplace transformation is given by the Bromwich integral

γ i∞ Í 1 1 º st

± ˆ ˆ ¯ ° ¯ ° µ

(IL.1) f t e f s ds

2πi γ i∞ ±

γ ˆ ° where is a vertical contour in the complex plane chosen such that all singularities of f ¯ s are to the left of it. As we have seen in Theorem L.2 the rule for inverse transforming a product is simply t Í 1

± ˆ ° ¯ ° ¯ · ° ² ¯ τ τ τ

(IL.2) f gˆ t f g t d 0

Finally we present a collection of used inverse Laplace–transformation rules from [S2]. Î

Í 1 1 1 ct

± ±

· · µ Ä µ§· Ð

(IL.3) 1 e s max 0 c

s c s ½

Í 1 1 γt 3 ±

γ ± 2

· · ¯ · °

(IL.4) s s 1 e t Æ

Æ 2 π Æ

Í 1 1 1 1 ± ± 2

(IL.5) t

s π

Æ Æ

Í 1 σt ±

± ˆ σ

¯ ° ¯ °

(IL.6) f s f t e ½

Í 1 d σt ±

± ˆ

° ¯ ° µ ¯ ° ¯ σ ψ (IL.7) s f s f t e if 0 0 ½ dt α

Í 1 1 Õ α 1 i Õ s

± 4t

(IL.8) e e

s πt

Æ Æ THE Ö –TRANSFORMATION 105

The Ë –Transformation

The main tool of this work is the Ë –transformation which is the discrete analogue of the

Laplace–transformation. The Ë –transformation can be applied to the solution of linear difference equations in order to reduce the solutions of such equations into those of algebraic equations in the complex z–plane. In this theses we used it to explicitly solve the ﬁnite difference schemes in the exterior domain in order to construct the discrete transparent boundary

conditions. The Ë –transformation is described in more detail in [S3], [S6], [S8]. It is deﬁned in the following way:

DEFINITION Z.1 ( Ë –transformation [S3]). The formal connection between a sequence and a complex function given by the correspondence ∞

Î n

ˆ ± Ë Ð ¯ ° µ Á µ ´ ´3Ä µ (Z.1) fn f z : ∑ fn z z CI z R fˆ

n ¼ 0

ˆ Î

¯ ° Ë Ð

is called Ë –transformation. The function f z is called –transformation of the sequence fn ,

µ µ?²§²§² À n 0 1 and R fˆ 0 denotes the radius of convergence. The discrete analogue of the Differentiation Theorem for the Laplace transformation is the

shifting theorem: Î

THEOREM Z.1 (Shifting Theorem [S3]). If the sequence fn Ð is exponentially bounded, i.e.

there exist C Ä 0 and c0 such that

Ô c0n

´ ´ µ µ µ§²§²§² µ

fn C e n 0 1

¯ °

then the Ë –transformation f z is given by the Laurent series (Z.1) and for the shifted sequence

Î Ð

gn with gn fn 1 holds º

Î ˆ Ë Ð ¯ °U· ²

(Z.2) fn 1 z f z z f0 º The initial values enter into the transformation of the shifted sequence. As a useful consequence of the shifting theorem we have:

Î ˆ Ë Ð ¯ ° ¯ °U· ²

(Z.3) fn 1 × fn z 1 f z z f0

º Î

Î n

Ð Ð µ µ§²?²§² Ó ∑

The convolution f g of two sequences f , g , n 0 1 is deﬁned by f g .

n n n n k ¼ 0 k n k ±

For the Ë –transformation of a convolution of two sequences we formulate the following theo-

rem: Î

ˆ Î ¯ ° Ë Ð ´ ´3Ä À ¯ ° Ë Ð ´ ´3Ä

THEOREM Z.2 ( [S8]). If f z fn exists for z R fˆ 0 and gˆ z gn for z

Ë Ó Ð ´ ´AÄ ¯ µ ° Rgˆ À 0, then there also exists fn gn for z max R fˆ Rgˆ with

Î ˆ

Ë Ó Ð ¯ ° ¯ ° ²

(Z.4) fn gn f z gˆ z Note that (Z.4) is nothing else but an expresssion for the Cauchy product of two power

series. We further need the Ë –transformation of a product of two sequences.

ˆ Î

¯ ° Ë Ð ´ ´4Ä À

THEOREM Z.3 (Complex Convolution Theorem [S8]). If f z fn exists for z R fˆ

Î Î

¯ ° Ë Ð ´ ´#Ä À Ë Ð ´ ´BÄ 0 and gˆ z gn for z Rgˆ 0, then there also exists fn gn for z R fˆ Rgˆ and the following relation holds:

Î 1 z 1

ˆ ±

Ð ¯ ξ ° ξ ² Ë ξ (Z.5) fn gn π f gˆ ξ d 2 i Ø

106 APPENDIX Ô

iϕ Ô

Å Åt´ ´Ú³ Ù ξ ϕ π

The integration path is the circle around the origin: r e , R fˆ r z Rgˆ, 0 2

Å Å ∞ (if Rgˆ 0 : R fˆ r ).

As a consequence one obtains immediately Plancherel’s theorem for the Ë –transformation:

ˆ Î

¯ ° Ë Ð ´ ´(Ä À

THEOREM Z.4 (Plancherel’s Theorem [S3]). If f z fn exists for z R fˆ 0 and

Î Î

¯ ° Ë Ð ´ ´=Ä À Å Ë Ð ´ ´=Ä gˆ z gn for z Rgˆ 0 with R fˆ Rgˆ 1. Then there also exists fn g¯n for z R fˆ Rgˆ and the following relation holds: ∞ 2π iϕ

Î 1 ˆ iϕ e ϕ Ë Ð¿¯ ° ¯ ° ²

(Z.6) ∑ fn g¯n fn g¯n z 1 f r e gˆ d 2π 0 r

n ¼ 0

Ù Å Å ³ Å Å ∞

The integration path is the circle deﬁned by R fˆ r 1 Rgˆ (if Rgˆ 0: R fˆ r ). Espe-

Å Å cially, if R fˆ 1, Rgˆ 1 then r 1 can be chosen to obtain: ∞ 1 2π

ˆ iϕ iϕ

¯ ° ¯ ° ϕ ²

(Z.7) ∑ fn g¯n f e gˆ e d 2π 0

n ¼ 0

¯ ° Ë Ë Ð¿¯ ° ¯ °

Note that gˆ z is not the –transformation of g¯n, but g¯n z gˆ z¯ holds.

The Inverse Ë –Transformation

Now we present the basic rules for calculating the inverse Ë –transformation which are es-

sential for formulating the discrete transparent boundary conditions.

Î Ð THEOREM Z.5 (Inverse Ë –transformation [S3]). If fn is an exponentially bounded se-

° Ë Ë quence and f ¯ z the corresponding –transformation then the inverse –transformation is given by

1 1 n 1 ±

ˆ ˆ ±

Ë ¯ ° ¯ ° µ µ µ§²§²?²4µ

(IZ.1) fn f z f z z dz n 0 1

2πi Ø

where Ù denotes a circle around the origin with sufﬁciently large radius.

ˆ 1

¯ ° Other inversion formulas can be obtained by using the fact that f z ± is a Taylor series or

ˆ ° if f ¯ z is a rational function of z, analytic at ∞ (cf. Section 1.3 in [S6]). The most important

formula is the inverse Ë –transformation of a product : n 1

± ˆ

Ë ¯ ° ¯ ° Ó µ

(IZ.2) f z gˆ z fn gn ∑ fk gn k ±

k ¼ 0 which is the discrete analogue of (IL.2).

ˆ Î

¯ ° Ë Ð

THEOREM Z.6 (Initial Value Theorem [S3]). If f z fn exists then

¯ ° ²

(IZ.3) f lim f z 0 ∞ z Ò

ˆ ° ∞ ¯ ∞

z can tend to on the real axis or on an arbitrary path, since f z is analytic at z .

ˆ ˆ ˆ 1 ¯ ° ¯ °U· ¯ °U· · This theorem, when repeatedly applied to f z , f z f0, f z f0 f1z ± , etc., provides

a method for the inversion of the Ë –transformation: n 1 n ± k

ˆ ±

¯ °· µ µ µ µ§²§²§² ² (IZ.4) f lim z f z ∑ f z n 0 1 2 n ∞ k z Ò

k ¼ 0 THE INVERSE Ö –TRANSFORMATION 107

Below we present some inverse Ë –transformation rules from [S3], [S6], [S8] that we used before:

1 Î 0

± δ Ë Ð µ (IZ.5) 1 n

1 1 1

Ë δ µ ´ ´AÄ µ

(IZ.6) z 1 z n

1 z n

Ë µ ´ ´3ÄI´ ´AÄ µ

(IZ.7) a z a 0

z · a

1 1 1 n 0

± ±

¯ · ° µ ´ ´AÄ*´ ´3Ä µ Ë δ (IZ.8) a a n z a 0

z · a

θ Å and for 0 Å 1 holds:

· · θ θ 1 1 1 1 0 1

± δ Ë · · · µ ´ ´3Ä µ

(IZ.9) n z

· ° ¯ · °

2 ¯ θz 1 θ 2 1 θ θ θ ½

· · · θ θ 1 z 1 1 0 1 1 0 1

± δ δ Ë · · µ ´ ´3Ä ²

(IZ.10) z

n ½ n

· ° ¯ · °

2 ¯ θz 1 θ 2θ 2θ 1 θ θ θ ½ From the generating function of the Legendre polynomials Pn [S4] we see that

1 z

Ë ¯ ° µ ´ ´3Ä ´ ´ÛµJ´ ´ µ (IZ.11) Pn µ z R max z1 z2 2

z · 2µz 1 ½

° ¯ °

λ ¯ λ ± holds and introducing the scaling variable we obtain for Pn µ : Pn µ λ

1 z 1 1

± ± ±

Ë ¯ ° µ ´ ´3Ä λ µ λ ² (IZ.12) Pn µ z R max z1 z2

λ 2 λ ° ·

¯ z 2µ z 1 ½

Here z1, z2 denote the zeros of the denominator in (IZ.11). 108 APPENDIX Bibliography

Programcodes

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Discrete Approach for Artiﬁcial BCs

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Difference Equations

[E1] S.N. Elaydi, An introduction to difference equations, Undergraduate Texts in Mathematics, Springer, New York, 1996. [E2] W. Gautschi, Computational aspects of three–term recurrence relations, SIAM Rev. 9 (1967), 24–82. [E3] R.E. Mickens, Difference Equations: Theory and Applications, 2nd ed., Van Nostrand Reinhold, New York, 1990. [E4] R.E. Mickens, Asymptotic properties of solutions to two discrete Airy equations, J. Difference Equ. Appl. 3 (1998), 231–239. [E5] R.E. Mickens, Asymptotic Solutions to a Discrete Airy equation, to be published in: J. Difference Equ. Appl. (2001).

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Finite Difference Schemes

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Mathematical Analysis

[M1] N. Dunford and J.T. Schwartz, Linear Operators, Part II: Self Adjont Operators in Hilbert Space, John Wiley Interscience Publishers, New York, 1963. [M2] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and Its Applications, 34. Cambridge Univ. Press, Cambridge, 1990. [M3] H. Heuser, Gewohnlic¨ he Differentialgleichungen: Einfuhrung¨ in Lehre und Gebrauch, Teubner, Stuttgart, 1989. [M4] L. Hormander¨ , The analysis of linear partial differential operators III: Pseudo–differential operators, Grundlehren der Mathematischen Wissenschaften 274, Springer–Verlag, Berlin, 1985. [M5] H.-O. Kreiss und J. Lorenz, Initial–Boundary Value Problems and the Navier–Stokes Equations, Academic Press, 1989. [M6] J.E. Lagnese, General boundary value problems for differential equations of Sobolev type, SIAM J. Math. Anal. 3 (1972), 105–119. [M7] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Part I: Functional Analysis, Academic Press, New York, 3rd ed. 1987.

Artiﬁcial Boundary Conditions for Parabolic Equations

[P1] E. Dubach, Artiﬁcial boundary conditions for diffusion equations: Numerical study, J. Comput. Appl. Math. 70 (1996), 127–144. [P2] M. Ehrhardt, Discrete Transparent Boundary Conditions for Parabolic Equations, Z. Angew. Math. Mech. 77 (1997) S2, 543–544. [P3] L. Halpern, Artiﬁcial BC’s for the Linear Advection Diffusion Equation, Math. Comp. 46 (1986), 425–438. [P4] L. Halpern and J. Rauch, Absorbing boundary conditions for diffusion equations, Num. Math. 71 (1995), 185–224. BIBLIOGRAPHY 111

[P5] J.-P. Loheac,´ An Artificial Boundary Condition for an Advection–Diffusion Problem, Math. Methods Appl. Sci. 14 (1991), 155–175. [P6] J.-P. Loheac,´ In– and Out–Flow Artificial Boundary Conditions for Advection–Diffusion Equations, Z. Angew. Math. Mech. 76 (1996) S5, 309–310. [P7] K. Wolter and A. Zisowsky, Numerical Solution of Second Order Fluid Stochastic Petri Nets with Reflecting and Transparent Boundary Conditions, submitted to: Proc. of the 11th Int. Conf. on Modell. Techn. and Tools for Comp. Performance Evaluation, Schaumburg, Illinois, USA (2000). Special Functions and Transformation

[S1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965. [S2] Bateman Manuscript Project, Tables of Integral Transforms, vol. 1, McGraw–Hill, 1954. [S3] G. Doetsch, Anleitung zum praktischen Gebrauch der Laplace-Transformation und der Z-Transformation, R. Oldenburg Verlag Munchen,¨ Wien, 3. Auﬂage 1967. [S4] N.N. Lebedev, Special Functions and their Applications, Selected Russian Publications in the Mathematical Sciences, Prentice–Hall, 1965. [S5] H.J. Nussbaumer, Fast Fourier transform and convolution algorithms, Springer Series in Information Sci- ences Vol. 2. Springer–Verlag, 1981. [S6] E.I. Jury, Theory and Application of the z-Transform Method, Wiley & Sons, New York, 1964. [S7] G. Szego,¨ Orthogonal Polynomials, Amer. Math. Soc. Colloquium Publications 23, 3rd ed. 1967. [S8] R. Vich, Z–Transformation, Theorie und Anwendung, Verlag Technik Berlin, 1964.

Transparent Boundary Conditions

[T1] A. Arnold, Numerically Absorbing Boundary Conditions for Quantum Evolution Equations, VLSI Design 6 (1998), 313–319. [T2] A. Arnold and M. Ehrhardt, A New Discrete Transparent Boundary Condition for Standard and Wide Angle “Parabolic” Equations in Underwater Acoustics, Proceedings of the “Third International Conference on Theoretical and Computational Acoustics”, Newark, New Jersey, USA, 1997, 623–635. [T3] A. Arnold and M. Ehrhardt, Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics, J. Comp. Phys. 145 (1998), 611–638. [T4] A. Arnold and M. Ehrhardt, Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations in Underwater Acoustics, Proceedings of the “Fourth European Conference on Underwater Acoustics”, Rome, Italy, 1998, 769–774. [T5] A. Arnold, Mathematical concepts of open quantum boundary conditions, to appear in: Transp. Theory Stat. Phys., 2000. [T6] V.A. Baskakov and A.V. Popov, Implementation of transparent boundaries for numerical solution of the Schroding¨ er equation, Wave Motion 14 (1991), 123–128. [T7] C.H. Bruneau and L. Di Menza, Conditions aux limites transparentes et artiﬁcielles pour l’equation´ de Schroding¨ er en dimension 1 d’espace, C. R. Acad. Sci. Paris, Ser. I 320 (1995), 89–94. [T8] T.W. Dawson, D.J. Thomson and G.H. Brooke, Non–local boundary conditions for acoustic PE predictions involving inhomogeneous layers, in 3rd European Conference on Underwater Acoustics, FORTH–IACM, Heraklion, 1996, pp. 183–188. [T9] L. Di Menza, Approximations numeriques´ d’equations´ de Schroding¨ er non lineair´ es et de modeles` associes´ , Ph.D. thesis, Universite´ Bordeaux I, 1995. [T10] J. Douglas Jr., J.E. Santos, D. Sheen and L.S. Bennethum, Frequency domain treatment of one–dimensional scalar waves, Math. Mod. and Meth. in Appl. Sc. 3 (1993), 171–194. [T11] M. Ehrhardt, Discrete Transparent Boundary Conditions for General Schroding¨ er–Type Equations, VLSI Design 9 (1999), 325–338. [T12] J.R. Hellums and W.R. Frensley, Non–Markovian open–system boundary conditions for the time–dependent Schroding¨ er equation, Phys. Rev. B 49 (1994), 2904–2906. 112 BIBLIOGRAPHY

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Underwater Acoustics and “Parabolic” Equation Models

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10. Nov. 1968 born in Berlin/Zehlendorf 08/1975–06/1981 Attendance at the Riemeister–Grundschule in Berlin 08/1981–06/1988 Attendance at the Werner-von-Siemens–Gymnasium in Berlin 06/1988 A–levels in mathematics and physics 10/1988–07/1995 Study of technomathematics at the Technical University in Berlin 04/1991 Partials ‘Vordiplom’ in technomathematics 09/1991–08/1995 Tutor at the Department of Mathematics of the Technical University in Berlin 08/1995 Master of Science ‘Diplom’ in technomathematics 09/1995-01/1999 Assistant at the Department of Mathematics of the Technical University in Berlin 02/1999–03/1999 TMR–scholarship at the University of Granada (Spain) 04/1999–09/1999 Assistant at the Department of Mathematics of the Technical University in Berlin since 10/1999 Assistant at the Department of Mathematics of the Saarland University

Saarbruck¨ en, Mai 2001

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