Discrete Artificial Boundary Conditions Vorgelegt Von Diplom
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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 mathe- matics. 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 bound- ary 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 evo- lution 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 absorb- ing 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 stabil- ity of the whole–space finite difference scheme. Especially for the Schrodinger¨ equa- tion of quantum mechanics the unconditional stability of the underlying