Convolution Quadrature for the Wave Equation with Impedance Boundary Conditions

a b, S.A. Sauter , M. Schanz ∗

aInstitut f¨urMathematik, University Zurich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland, e-mail: [email protected] bInstitute of Applied Mechanics, Graz University of Technology, 8010 Graz, Austria, e-mail: [email protected]

Abstract

We consider the numerical solution of the wave equation with impedance boundary conditions and start from a boundary integral formulation for its discretization. We develop the generalized convolution quadrature (gCQ) to solve the arising acoustic retarded potential integral equation for this impedance problem. For the special case of scattering from a spherical object, we derive rep- resentations of analytic solutions which allow to investigate the effect of the impedance coefficient on the acoustic pressure analytically. We have performed systematic numerical experiments to study the convergence rates as well as the sensitivity of the acoustic pressure from the impedance coefficients. Finally, we apply this method to simulate the acoustic pressure in a building with a fairly complicated geometry and to study the influence of the impedance coefficient also in this situation. Keywords: generalized Convolution quadrature, impedance boundary condition, boundary integral equation, time domain, retarded potentials

1. Introduction

The eﬃcient and reliable simulation of scattered waves in unbounded exterior domains is a numerical challenge and the development of fast numerical methods

∗Corresponding author

Preprint submitted to Computer Methods in Applied Mechanics and EngineeringAugust 18, 2016 is far from being matured. We are interested in boundary integral formulations

5 of the problem to avoid the use of an artiﬁcial boundary with approximate transmission conditions [1, 2, 3, 4, 5] and to allow for recasting the problem (under certain assumptions which will be detailed later) as an integral equation on the surface of the scatterer. The methods for solving the arising integral equations can be split into a)

10 frequency domain methods where an incident plane wave at prescribed frequency excites a scattered ﬁeld and a time periodic ansatz reduces the problem to a purely spatial Helmholtz equation and b) time-domain methods where the excitation is allowed to have a broad temporal band width and, possibly, an a-periodic behavior with respect to time.

15 For the solution, an ansatz as an acoustic retarded potential integral equation (RPIE) is employed. Among the most popular methods for discretizing this equation are: a) the convolution quadrature (CQ) method [6, 7, 8, 9, 10, 11] and b) the direct space-time Galerkin discretization (see, e.g., [12, 13, 14, 15, 16, 17]). In this paper, the generalized convolution quadrature (gCQ) is considered

20 for the discretization of the RPIE. This method has been introduced in [9, 18] for the implicit Euler time method and for the Runge-Kutta method in [19]. In contrast to the original CQ method the gCQ method allows for variable time stepping. We apply this method to the wave equation with linear impedance bound-

25 ary condition (for non-linear boundary conditions we refer to [20, 21, 22]) and study the effect of different values of the impedance coefficient on the solution analytically for a spherical scatterer and numerically for concrete applications with a fairly complicated geometry. The paper is organized as follows. In Section 2, we will introduce the wave

30 equation with impedance conditions and the corresponding retarded potential integral equation. In Section 3, we introduce the generalized convolution quadrature method for the RPIE with impedance boundary conditions. New represen- tations for analytic solutions in the case of a spherical scatterer are derived in Section 4 which allow for a stable numerical evaluation.

2 35 Numerical experiments are described in Section 5. First, systematic studies of the convergence order have been performed for problems where the exact solution is known and the effect of the impedance coefficient on the acoustic pressure is investigated numerically. Then, the method is applied to model the effect of the impedance coefficient for the acoustic pressure in the atrium of the

40 “Institut für Mathematik” at the University Zurich. In 2010/11 an acoustic absorber has been installed at the ceilings to improve the acoustics in the building. Our goal is to model this effect numerically by the gCQ method and the results are also described in Section 5. In the Conclusions 6 we summarize the main findings in this papers.

45 2. Setting

3 Let Ω − R be a bounded Lipschitz domain with boundary Γ := ∂Ω and let ⊂ Ω+ := R3 Ω denote its unbounded complement. Let n denote the unit normal \ − vector to Γ pointing in the exterior domain Ω +. We consider the homogeneous wave equation (with constant sound speed c in the medium) for σ +, ∈ { −} ∂ u c2∆u =0 inΩ σ R , tt − × >0 σ u (x, 0) = ∂tu (x, 0) = 0 in Ω , (1) γσ (u) σ α γσ (∂ u) = f (x, t ) on Γ R , 1 − c 0 t × >0 σ where γ1 = ∂/∂n is the normal derivative applied to a suﬃciently smooth σ σ function in Ω and γ0 denotes the trace operator to Γ applied to a suﬃciently σ + smooth function in Ω . If the domain Ω Ω−, Ω is clear from the context ∈ { } we skip the superscript σ and simply write γ1 and γ0. In (1), α denotes the

50 non-negative admittance, which is the inverse of the speciﬁc impedance function of the surface Γ. Speciﬁc means that the impedance is scaled by the density and the wave velocity. The value of α is mathematically non-negative, however, realistic values are in the range 0 α 1. The lower limit models a sound ≤ ≤ hard wall and the upper limit is a totally absorbing surface. Further, measured

55 values show a frequency dependence and are listed in national regulations like the ONORM¨ in Austria ( ONORM¨ EN 12354-6).

3 Such kind of absorbing boundary condition is a simple possible choice to model the absorption of a surface. Certainly, more complicated models exist which takes higher derivatives into account. The most realistic models consider

60 an absorbing layer of porous material on the real surface, which is the computational most expensive way (see, e.g., [23, 24]). Here, this simple model is used as it is common in real world applications.

2.1. Layer Potentials We employ layer potentials to express the solution in terms of retarded potentials (cf. [25, 26, 12, 27]). The ansatz for the solution u as a single layer potential is given by

x y ϕ y, t k −c k u (x, t ) = ( ϕ) ( x, t ) := − dΓ (x, t ) Ωσ R , S ∗ 4π x y y ∀ ∈ × >0 ZΓ k − k which satisﬁes the ﬁrst two equations in (1). The density ϕ then is determined via the third equation. Alternatively we can represent the solution as a double layer potential

x y ∂ ψ z, t k −c k u (x, t ) = ( ψ) ( x, t ) := − dΓ D ∗ ∂n (y) 4π x y  y ZΓ k − k z=y  x y  ψ y, t k − k 1 n (y) , x y − c 1 ˙ x y = h −2 i + ψ y, t k − k dΓy. 4π Γ x y  x y c − c  Z k − k k − k   The application of the trace γ0 and normal trace γ1 to u involves the following boundary integral operators

x y ϕ y, t k −c k ( ϕ) ( x, t ) = − dΓ , V ∗ 4π x y y ZΓ k − k x y x y ψ y, t k − k ψ˙ y, t k − k 1 n (y) , x y − c − c ( ψ) ( x, t ) = h −2 i + dΓy, K ∗ 4π Γ x y  x y c  Z k − k k − k  x y x y  ϕ y, t k − k ϕ˙ y, t k − k 1 n (x) , y x − c − c ( ′ ϕ) ( x, t ) = h −2 i + dΓy, K ∗ 4π Γ x y  x y c  Z k − k k − k ∂   ( ψ) ( x, t ) = ( ψ) ( x, t ) W ∗ −∂n (x) D ∗

4 for almost all ( x, t ) Γ R , more precisely, for all ( x, t ) Γ R , where Γ ∈ × >0 ∈ × >0 is smooth in a neighborhood of x. Then, it holds for σ +, ∈ { −}

γσ ( ϕ) = ( ϕ) , 0 S ∗ V ∗ σ ϕ γ ( ϕ) = σ ′ ϕ , 1 S ∗ − 2 − K ∗ ψ γσ ( ψ) = σ + ψ 0 D ∗ 2 K ∗ γσ ( ψ) = ψ, − 1 D ∗ W ∗

where, again, these equations hold almost everywhere on Γ R . × >0 The third equation in (1) leads to the boundary integral equation for the single layer ansatz

ϕ α σ ′ ϕ σ ( ϕ˙) = f a.e. in Γ R . (2) − 2 − K ∗ − c V ∗ × >0 The double layer ansatz leads to the boundary integral equation

σα ψ˙ ψ σ + ψ˙ = f a.e. in Γ R>0. (3) −W ∗ − c 2 K ∗ ! ×

65 3. Generalized Convolution Quadrature (gCQ)

The convolution quadrature method has been developed by Lubich, see [6, 7, 28, 29, 30] for parabolic and hyperbolic problems. The idea is to express the RPIE as the inverse Laplace transform applied to the counterpart of the RPIE in the Fourier-Laplace domain which reduces the problem to the solution of a scalar

70 ODE of the form y′ = sy + g, for s being the variable in the Laplace domain. The temporal discretization then is based on the numerical approximation of the solution of this ODE by some time-stepping method and the transformation of the resulting equation back to the original time domain. The original CQ method requires constant time stepping. However, if the

75 right-hand side is not uniformly smooth and/or contains non-uniformly dis- tributed variations in time, and/or consists of localized pulses, the use of adaptive time stepping becomes very important in order to keep the number of time

5 steps reasonably small. The generalized convolution quadrature (gCQ) has been introduced in [9, 18, 19] and allows for variable time stepping. We recall the deﬁnition of the Laplace transform and its inverse

∞ st fˆ(s) := (f) ( s) := e− f (t) dt L Z0 1 f (t) := est fˆ(s) ds for γ := ρ + i R and some ρ > 0 2π i Zγ for a Laplace-transformable function f, where the vertical contour γ runs from ρ i to ρ + i . The Laplace transformed boundary integral operators are − ∞ ∞ given for sufficiently smooth functions Φ , Ψ : Γ C by (cf. [25], [27]) → s x y /c e− k − k σ (s) Φ ( x) = Φ ( y) dΓy x Ω s Cρ, S Γ 4π x y ∀ ∈ ∀ ∈ Z sk x−y /ck e− k − k n (y) , x y 1 s σ C b (s) Ψ ( x) = h 2 − i + Ψ ( y) dΓy x Ω s ρ, D Γ 4π x y x y c ∀ ∈ ∀ ∈ Z s x y /ck − k k − k e− k − k b(s) Φ ( x) = Φ ( y) dΓy x Γ s Cρ, V Γ 4π x y ∀ ∈ ∀ ∈ Z sk x −y /ck e− k − k n (y) , x y 1 s C b (s) Ψ ( x) = h 2 − i + Ψ ( y) dΓy x Γ s ρ, K Γ 4π x y x y c ∀ ∈ ∀ ∈ Z s x y /ck − k k − k b e− k − k n (x) , y x 1 s C ′ (s) Φ ( x) = h 2 − i + Φ ( y) dΓy x Γ s ρ, K Γ 4π x y x y c ∀ ∈ ∀ ∈ Z∂ k − k k − k c (s) Ψ = (s) Ψ ( x) x Γ s C , W −∂n (x)D ∀ ∈ ∀ ∈ ρ wherec b C := s C Re s ρ for some ρ > 0. (4) ρ { ∈ | ≥ } The definition of the (generalized) convolution quadrature depends of the growth behavior of the inverse Laplace transformed integral operator with respect to the frequency variable. This operator must decay fast enough such that the integral over the infinite contour γ exists. In our case this requires a regularization parameter µ N which will be specified later. We denote by f (µ) as usual the ∈ 0 µ-th derivative of f. For µ Z, we define ∈

µ (s) := s− (s) (5) Vµ V

80 while , , and other frequencyb dependentb integral operators are deﬁned Kµ K′µ Wµ analogously. b c c

6 The application of the inverse Laplace transform to (2) leads to the following integro-diﬀerential equation in the frequency domain 1 Qσ (s) U (s, t ) ds = f (µ), (6a) 2π i γ µ Z − ∂tU (s, t )c = sU (s, t ) + Φ , U (s, 0) = 0 , (6b) σ α with Qσ (s) := I (s) σ s (s) , (6c) − 2 − K′ − c V where we have suppressedc the x-dependencec of the functionsb and operators in the notation. The same technique can be applied to (3) and we obtain 1 Rσ (s) V (s, t ) ds = f (µ) (7a) 2π i γ µ Z − ∂tV (s, t )c = sV (s, t ) + Ψ , V (s, 0) = 0 , (7b) α 1 with Rσ (s) := (s) s I + σ (s) . (7c) −W − c 2 K The operators Qσ (cs) and Rσ (sc) are continuous andb invertible in appropriate µ 85 Sobolev spaces H (Γ) on the surface Γ and have algebraic growth behavior with c c respect to s . Since the growth exponent will be a control parameter for the | | generalized convolution quadrature, we provide here the relevant theorem.

Theorem 3.1. Let ρ > 0 in (4) and ρ := min 1, ρ . Let the admittance 1 { } function satisfy

0 < α min := min α (x) max α (x) =: αmax < . x Γ ≤ x Γ ∞ ∈ ∈ Then, the operators Qσ (s) and Rσ (s) and their inverses satisfy the following continuity estimates for all s C c ∈ cρ σ αmax 2 Q (s) C1 C2 + s , (8a) H−1/2(Γ) H−1/2(Γ) ≤ c | | ← σ αmax 5/2 cR (s ) C1 C2 + s , (8b) H−1/2(Γ) H1/2(Γ) ≤ c | | ←

and c 1 σ− c Q (s) − − C1 s , (9a) H 1/2(Γ) H 1/2(Γ) ≤ αmin | | ← 1 σ− 2 cR (s ) C2 s . (9b) H−1/2(Γ) H1/2(Γ) ≤ | | ←

c 7 The constants C1, C2 only depend on ρ and ρ1.

Proof. The operator Qσ (s) and Rσ (s) can be expressed in terms of the Dirichlet- to-Neumann map DtN σ and the Neumann-to-Dirichlet map NtD σ for the exte- c c rior ( σ = 1) and the interior ( σ = 1) domain Ω σ. It holds (cf. [31, Appendix − 2])

α Qσ (s) = σ σ DtN σ + sI (s) , − − c V α Rcσ (s) = I s σ NtD σ b(s) . − − c W For the continuity estimatesc we employ the representationsc (6c) and (7c) and well known continuity estimates for , , , (see, e.g., [12, Prop. 3], [32, K K′ V W formulae (10), (11)], [31, Appendix 2]) to obtain b c b c 1 s 3/2 α s 2 Qσ (s) + C | | + C max | | , −1/2 −1/2 3/2 2 H (Γ) H (Γ) ≤ 2 ρρ c ρρ 1 ← 1 2 3/2 c s αmax 1 s Rσ (s) C | | + s + C | | − 3/2 H 1/2(Γ) H1/2(Γ) ≤ ρρ 1 c | | 2 ρρ ! ← 1 c for all s C . From this, estimates (8) follow by using s Re s ρ. ∈ ρ | | ≥ ≥ For the inverses we employ [31, Prop. 17, 18]:

2 i Arg s σ α ρρ 1 αmin 2 1/2 Re e− Φ, σ DtN + sI Φ C + s Φ H1/2(Γ) Φ H (Γ) , − c L2(Γ) ≥ s c | | k k ∀ ∈ | | i Arg s α σ αmin 1 2 1/2 Re e Ψ, I s σ NtD Ψ Re s + C ρρ 1 Ψ H−1/2(Γ) Ψ H− (Γ) . − c L2(Γ) ≥ c s k k ∀ ∈ | | The Lax-Milgram lemma implies

1 1 The combination with well known mapping properties of − and − (cf. [31, V W b c

8 Prop. 16, 19]) leads to

1 1 σ 1 σ α − Q − (s) − (s) σ DtN + sI −1/2 −1/2 −1/2 1/2 H (Γ) H (Γ) ≤ V H (Γ) H (Γ) − c −1/2 1/2 ← ← H (Γ) H (Γ) ← c bs 2 s C , | | | α| 2 ≤ ρρ 1 Cρρ 2 + min s 1 c | | 1 1 σ 1 α σ − R − (s) − (s) I s σ NtD −1/2 1/2 1/2 −1/2 H (Γ) H (Γ) ≤ W H (Γ) H (Γ) − c −1/2 −1/2 ← ← H (Γ) H (Γ) ← c cs s C | |2 αmin | | ≤ ρρ 1 C c ρρ 1 + Re s 90 from which the estimates of the inverse operator follow.

Remark 3.2. According to the growth of the inverse Laplace transformed inte- 1 1 gral operators Qσ− (s), Rσ− (s) we deﬁne

3 for problem (2), c µc:=  4 for problem (3).  1 1 This deﬁnition implies that the contour integrals Qσ− (s) ds and Rσ− (s) ds γ µ γ µ exist. R R c c By approximating the ODE in (6b) and (7b), by a time stepping scheme and

replacing the integral γ . . . by a contour quadrature leads to the approximation of (2) and (3) by generalizedR convolution quadrature. The following algorithm is taken from [18] and employs the implicit Euler method for discretizing the ODEs in (6b), (7b); for a generalization to Runge-Kutta methods we refer to N [19]. Let time steps ( tj)j=0 be given

0 = t0 < t 1 <...

and introduce the corresponding mesh sizes ∆ j = tj tj 1 The implicit Euler − − method for solving (6b) deﬁnes approximations U (s) U (s, t ) by n ≈ n Un 1 ∆nΦn U = − + , U = 0 . n 1 s∆ 1 s∆ 0 − n − n Inserting this into (6a) at time tn and using Cauchy’s integral theorem results in Qσ (s) σ 1 (µ) 1 µ Q Φn = f (tn) − Un 1 (s) ds. (10) µ ∆n − 2π i 1 s∆n − − ZC c− c 9 1 N Here, is a bounded contour which encircles all poles ∆− of the integrand C m m=1 and is clockwise oriented (cf. [18]). #

95 In [33, 18] a quadrature rule for the contour integral in (10) has been developed and analyzed. Denote by s the nodes and by w the weights, ℓ ∈ C ℓ ℓ = 1 , , N Q. By replacing the integral . . . by the quadrature formula, we · · · C can formulate the generalized convolutionR quadrature in an algorithmic way.

Deﬁnition 3.3 (gCQ) . The generalized convolution quadrature for solving (6), (7) is given by the procedure: For n = 1 , . . . , N do

0 n = 1 , Un 1 (sℓ) := ℓ = 1 , . . . N Q −  Un−2(sℓ) ∆n−1  1 ∆ − s + 1 ∆ − s Λn 1 n 2  − n 1 ℓ − n 1 ℓ − ≥    (11a)

NQ 1 (µ) Z (sℓ) Z Λn := f (tn) wℓ Un 1 (sℓ) ds, (11b) ∆n − 1 sℓ∆n − Xℓ=1 −

σ where Λn := Φ n and Z := Q for problem (6) and Λn := Ψ n and Z := µ − 100 Rσ for problem (7). µ c − c 4. Analytic Solutions

In this section, we provide sample solutions for the wave equation with impedance boundary conditions on the sphere. This allows to study explicitly the inﬂuence on the admittance function α on the acoustic pressure and to

105 compare the numerical solution with an exact solution. There are essentially two diﬀerent ways for the construction of exact refer- ence solutions for boundary integral equations. One way is to employ Kirchhoﬀ’s formulae (see, e.g., [27]) for the wave equation (2), (3) in the Laplace domain which can be combined such that we get the relations

+ + + + Q− (s) γ1 u = R− (s) γ0 u , + + Q (s) γ1−u− = R (s) γ0−u− d b c b d b c b

10 + valid for homogeneous interior and exterior solutions u−, u of the wave equation in the Laplace domain. Such solutions can be created by a source distribution located outside the computational domain

s x y /c s x y /c e− k − k + + e− k − k u− (x) := for fixed y Ω and u (x) := for fixed y Ω− 4π x y ∈ 4π x y ∈ k − k k − k and allow to derive sample solutions for all equations Qσ (s) Φ = f and Rσ (s) Ψ = f, where f is then defined as the application of one of the integral operators c b c Qσ, Rσ to the trace or normal trace of u. b b Another approach can be applied for the sphere since the eigenpairs of the c c b 110 boundary integral operator for the acoustic single layer operator are known. From this, we will derive the eigensolutions of the time-space integral equation for the retarded acoustic single layer potential. In this case, the right-hand side is not defined as the application of an integral operator to a trace but known explicitly. In addition, this approach allows to study the behavior of the solution

115 for higher eigenmodes and regularity issues although this study is beyond the scope of this paper. For pure Dirichlet and Neumann problems, such sample solutions have been derived, e.g., in [34] and [35]. Let Ω R3 denote the unit ball with surface Γ := S . Let Y m denote the ⊂ 2 n spherical harmonics. We assume that the right-hand side f is given by

m f (x, t ) := f (t) Yn (12)

with a slight abuse of notation. Note that the spherical harmonics are eigen- functions of the boundary integral operators , , , , i.e., V K K′ W

m (Z) s m ZY = λ Y for Z b b c, ,c′, . n n c n ∈ {V K K W} b

11 Explicitly it holds (cf. [36, 37]) 1

( ) (1) ( ) 1 2 (1) λnV (s) = sj (i s) hn (i s) , λ nK (s) = i s j (i s) ∂h n (i s) , − n 2 − n

′ ( ) 3 (1) ( ) ( ) λnW (s) = s ∂j (i s) ∂h n (i s) , λ nK (s) = λnK (s) , − n (1) with the spherical Bessel and Hankel functions j , hn (cf. [38, 10.4.7]) and n § (1) ∂j n, ∂h n denoting their ﬁrst derivatives. Then the Laplace transformed equa- m m tions (2), (3) take the form (by using the ansatz ϕ = ϕ (t) Yn and ψ = ψ (t) Yn )

1 s σ ( ) ( ) ϕ = η− α, f (s) with η (α, s ) := λnK (s) σαsλ nV (s) , n n c n − 2 − − 1 s ( ) 1 ( ) ψn = γ− #α, f (s) with γn (α, s ) := λnW (s) αs + σλ nK (s) . c n c b − − 2 # (13) c b

4.1. The Case n = 0

For n = 0, we have

− − ( ) 1 e 2s ( ) 1 e 2s 1 V K λ0 (s) = −2s , λ 0 (s) = 2 + 2s− (s + 1)

− ′ ( ) s 1+e 2s(1+ s) ( ) ( ) W K K λ0 (s) = (1 + s) − 2s , λ 0 (s) = λn (s) .

1Note that in [37, (2.6.116)] is a misprint. For the Wronskian W of the spherical Bessel (1) functions jℓ, hℓ , it holds (1) i W h (z) , j ℓ (z) = − ℓ z2 so that in [37, (3.2.22)] on the right-hand side a factor i is missing. By the same reason a factor i is also missing in the formulae [37, (3.2.23), (3.2.24)], while in formula [37, (3.2.26)] a factor i is missing only in front of the first term in the bracket. Note also that our definition of W differs from the definition [37, (3.2.17)] by a sign.

12 1 1 For the reciprocal symbols η0− , γ0− we obtain

1 2s η− (α, s ) := 0 −σs (1 + α) + 1 e 2s (s (1 + σα ) + 1) − − ℓ 2s ∞ (s (1 + σα ) + 1) e 2s = − , −σs (1 + α) + 1 σs (1 + α) + 1 Xℓ=0 1 2s γ− (α, s ) := 0 −(1 + α) s2 σαs 1 + e 2s (s + 1) ( s (σα + 1) + 1) − − − ℓ 2s ∞ (s + 1) (( σα + 1) s + 1) e 2s = ( 1) ℓ − . −(1 + α) s2 σαs 1 − (1 + α) s2 σαs 1 − − Xℓ=0 − − 4.1.1. Single Layer Ansatz, exterior problem

120 We restrict for the analytic considerations to the exterior problem and to formulation (2). For the exterior problem σ = +1, we get

1 2s ∞ 2ℓs η− (α, s ) = e− . 0 −s (1 + α) + 1 Xℓ=0 The inverse Laplace transform of this function is given by [39, 4.1(4) with a = 1 and b = 2 ℓ]

1 1 ∞ 1 2 − η− (α, ) (t) = H (t 2ℓ) − • (t 2ℓ) . L 0 • − − L (1 + α) + 1 − ℓ=0 • # X We have

t 1+ α 1 2 2 1 1 1 [39, 5.2(1)] 2 e− − • (t) = − 1 (t) = δ0 (t) L (1 + α) + 1 1 + αL − 1 + α s + 1 1 + α − 1 + α • 1+ α ! !

with the Dirac delta distribution δ0 so that

t−2ℓ 1+ α 1 1 2 ∞ e− − η− (α, ) (t) = H (t 2ℓ) δ (t 2ℓ) . (14) L 0 • −1 + α − 0 − − 1 + α ℓ=0 ! # X Hence, the density for the single layer ansatz for the exterior problem is given

13 by 2

ct/ 2 ⌊ ⌋ t 2ℓ/c − − + 2 2ℓ c − c(t τ) 2ℓ ϕ (t) = f t e− 1+ α f (τ) dτ −1 + α − c − 1 + α ℓ=0 Z0 ! X (15) with the notation x for the largest integer less than or equal to x. ⌊ ⌋ Example 4.1 (bump functions) .

125 a. General bump function. For ρ > 1 and υ > 0, we choose f in (2) as υ cρt cρt 1+ α the bump function fυ (t) := 1+ α e− . Then, the density in (15) can be written in the form 3

ct/ 2 υ υ ⌊ ⌋ + 2ρ ct 2ℓ ρ (ct 2ℓ) ϕ (t) = − e− 1+ α − (17) −1 + α 1 + α Xℓ=0 (ρ 1) − ct −2ℓ γ υ + 1 , 1+ α (ct 2ℓ) e− 1+ α − . − (ρ 1) υ+1  −  υ ct b. For fυ (t) = ( ct ) e− it holds

ct/ 2 ⌊ ⌋ + 2 υ (ct 2ℓ) ϕ (t) = (ct 2ℓ) e− − (18) −1 + α − ℓ=0 υ X (1 + α) α ct −2ℓ γ υ + 1 , (ct 2ℓ) e− 1+ α . − αυ+1 1 + α −

· 2Let κ > 0. Then L−1 fˆ (t) = κ L−1 (f) (κt ) (see [39, 4.1(4) with a = κ, b = 0.]). κ − ! 1 a ! Also note that R δ0 (κt a) = κ δ0 κ . 3For r > 0, µR > 0, β ∈ R it holds !

r − γ (µ + 1 , rβ ) sµ e βs ds = , (16) µ+1 Z0 β where z − − γ (a, z ) := ta 1e tdt Z0 is an incomplete Gamma function (cf. [38, 8.2.1]).

14 4.2. The Solution of the Wave Equation in Ωσ

The Laplace transformed solution of the boundary integral equation for the single layer operator (13) with right-hand side as in (12) is given by

1 s m ϕ (x, s ) := η− α, f (s) Y (x) . n n c n This leads to the solutionb of the interior and exteriorb Laplace transformed wave equation (1)

s x y /c σ 1 s e− k − k m σ R u (x, s ) = (s) ϕn (x, s ) := ηn− α, f (s) Yn dΓy (x, t ) Ω >0 S c Γ 4π x y ∀ ∈ × Z k − k andb we haveb to evaluateb the applicationb of the Laplace transformed single layer m potential to the spherical harmonics Yn . Let

U m := (s) Y m. n S n m b b Then Un satisﬁes the transmission problem s 2 b ∆U m + U m =0 in R3 Γ, − n c n \ m γ0Un = 0 b Γ b h m i m γ1Un = Yn b Γ − mh i ∂Un s m 1 + b Un = o x − x . ∂r c k k k k → ∞ b b From [37, (2.6.53), (2.6.55)] we conclude that the solution in spherical coordi- nates x = rζ has the form

m m βn− (s) jn (i sr ) 0 r < 1 Un (x, s ) = Yn (ζ) ≤  β+ (s) h(1) (i sr ) r > 1  n n b with coeﬃcients βn± to be determined via the transmission conditions:

+ (1) + (1) ′ β− (s) j (i s) = β (s) h (i s) and i sβ − (s) j′ (i s) i sβ (s) h (i s) = 1 . n n n n n n − n n By solving this for βn± we end up with

(1) s hn (i s) jn (i sr ) 0 r 1, U m (x, s ) = ρ , r Y m (ζ) with ρ (s, r ) := s ≤ ≤ n n c n n  (1) − j (i s) hn (i sr ) 1 r.  n ≤ b  15 This leads to

σ 1 s s m σ u (x, s ) = η− α, f (s) ρ , r Y (ζ) (rζ, t ) Ω R n c n c n ∀ ∈ × >0 For nb= m = 0, we get b

s r 1 s(r+1) e− | − | e− ρ (s, r ) := − 0 2sr so that s uσ (x, s ) = q α, , r f (s) Y 0 0 c 0 with b b s r 1 s(r+1) 2s ℓ ρ0 (s, r ) 1 e− | − | e− ∞ (s (1 + σα ) + 1) e − q0 (α, s, r ) := = − . η0 (α, s ) − r σs (1 + α) + 1 σs (1 + α) + 1 Xℓ=0 4.2.1. The Solution of the Wave Equation in Ω+ For σ = +1, we get

1 1 ∞ s(2 ℓ+r 1) s(2 ℓ+r+1) q (α, s, r ) = e− − e− . 0 − r s (1 + α) + 1 − Xℓ=0 The inverse Laplace transform applied to u+ can be computed by similar tech- niques as used for (14) b − − 1 1 1 ∞ t (2 ℓ+r 1) − (q (α, , r )) ( t) = H (t (2 ℓ + r 1)) e − 1+ α L 0 • − r 1 + α − − Xℓ=0 t−(2 ℓ+r+1) H (t (2 ℓ + r + 1)) e − 1+ α − − and the exterior solution u+ ﬁnally is given by

− ct r+1 − 2 2ℓ+r 1 t c − − − + c ⌊ ⌋ − c(t τ) (2 ℓ+r 1) u (rζ, t ) = e− 1+ α f (τ) dτ −2√π (1 + α) r  ℓ=0 Z0  X ct −r−1  2 t 2ℓ+r+1 ⌊ ⌋ − c c(t−τ)−(2 ℓ+r+1) e− 1+ α f (τ) dτ  − 0 Xℓ=0 Z −  t r 1  c ct −r+1 − c cτ = e− 1+ α e 1+ α f (τ) dτ, (19) −2√π (1 + α) r Z0 0 1 130 where we used Y0 = (2 √π)− .

16 Example 4.2 (bump functions (revisted)) . Let fυ (t) be the general bump function as in Example 4.1 and, for r > 1, we deﬁne τ := ct (r 1) . Let − − (τ) := max 0, τ . Then + { } τ υ 1+ α + ρ ρ 1 e− u (rζ, t ) = γ υ + 1 , − τ+ −2√π (ρ 1) υ+1 1 + α r − υ ct + For f (t) = ( ct ) e− the representation of u simpliﬁes to

υ τ (1 + α) α e− 1+ α u+ (rζ, t ) = γ υ + 1 , τ . (20) −2√πα υ+1 1 + α + r The dependence of u+ (rζ, t ) with respect to α 0 is smooth. For r 1, it ≥ ≥ holds υ+1 τ 1+ α + 1 (τ)+ e− u (rζ, t ) α . ≤ √π (υ + 1) (1 + α) 1 + 1+ α τ r

1 With increasing α the amplitude of the acoustic pressure is damped by 1+ α and the same holds for the exponent which determines the reverberation time.

Proof. By using (19) and the deﬁnition of fυ we get

r−1 υ − t c − + c ct r+1 − c(ρ 1) τ cρτ u (rζ, t ) = e− 1+ α e− 1+ α dτ. −2√π (1 + α) r 1 + α Z0 1st Case: ct r 1. ≤ − Obviously u+ = 0 in this case. 2nd Case: r 1 < ct . We employ (16) and obtain − ct r+1 υ − γ υ + 1 , (ρ 1) − + ρ ct r+1 1+ α u (x, t ) = e− 1+ α − . (21) −(ρ 1) υ+1 2√πr − υ ct Next we set ρ = 1 + α so that the right-hand side becomes f (t) = ( ct ) e− and we obtain

r 1 0 t − , + ζ ≤ c u (r , t ) = υ γ υ+1 , α (ct r+1) ct −r+1  (1+ α) ( 1+ α ) 1+ α r 1 υ+1 − e− t > − .  − α 2√πr c From [38, 8.10.2 with a > 1 therein] we conclude that for a 1 and x 0 it ≥ ≥ holds a 1 a x − x 2x γ (a, x ) 1 e− . | | ≤ a − ≤ a (x + 1) 17 For τ = ct (r 1) and r > 1, we have − − υ+1 τ 1+ α + 1 (τ)+ e− u (rζ, t ) α . ≤ √π (υ + 1) (1 + α) 1 + 1+ α τ r

135

5. Numerical Experiments

The purpose of this Section is to show a) how the proposed algorithm per- forms for model problems and b) its applicability to real-world problems. For the ﬁrst goal, a systematic convergence study is presented, which utilizes the

140 analytical solution for the sphere from Section 4. To show the applicability for real-world applications we computed the sound pressure field in the atrium of the “Institut für Mathematik” at the University Zurich with gCQ and the influence of sound absorbing material. All computations are done in 3-D and a classical matrix-oriented spatial

145 boundary element discretization is employed for a systematic study of the behavior of the gCQ. All regular integrals are performed with Gaussian quadrature formulas. The singular integrals are treated with the formulas as in [40]. The geometrical discretization is done with linear triangles and the data are approx- imated by piecewise linear shape functions. For the solution a direct solver is

150 used. All implementations are done within the BE-library HyENA [41].

5.1. Unit sphere: Convergence and inﬂuence of α The geometry chosen is a sphere with radius 1 m with a coordinate system ﬁxed at the midpoint. The scattering into the outer air is considered and the respective analytical solutions can be found in Section 4. For the spherical harmonics in the right-hand side the case of (12), we choose n = m = 0. The time behavior of the right-hand side is the discussed bump function

υ ct fυ (t) = ( ct ) e− ,

with υ > 0, i.e., the analytical solutions can be found in (18) for the density and in (20) for the pressure. For all tests the material data from air are used, i.e., c =

18 343 m/s is set. In contrast to the classical CQ method, the gCQ method allows

155 for a variable step size which becomes important to approximate solutions with singular behavior, e.g., at the initial or later times. Note that for υ R N ∈ >0\ υ , υ the bump function f is H¨older continuous, more precisely f C⌊ ⌋ { } with υ υ ∈ υ denoting the integer part of υ and υ := υ υ . From (18) it follows that ⌊ ⌋ { } − ⌊ ⌋ the non-smooth behavior of fυ at t = 0 inherits qualitatively the same order of

160 non-smoothness to the density function at time points t = 2 ℓ/c , ℓ N , i.e., ℓ ∈ 0 + υ , υ ϕ C⌊ ⌋ { }. Since u is deﬁned via an integration (involved in the gamma ∈ + υ +1 , υ function γ, see (20)) we conclude that u C⌊ ⌋ { }. + ∈ For simplicity, we assume υ (0 , 1). If we want to distribute N time points ∈ in the interval [0 , T ] such that a piecewise constant interpolation converges as 1 N − , the choice O n χ t = T , n = 0 , . . . , N with grading exponent χ = 1 /υ (22) n N of the mesh points is recommended. Since we employ the BDF 1 method for the time discretization we expect an error in the approximation of the density

165 function of (∆ t ) for ∆ t = T/N . For a uniform mesh with t = O const const n n∆t we expect a reduced convergence order of ((∆ t )υ). To achieve const O const 1 a comparable error of N − for a constant mesh width one has to choose a O 1/υ constant time mesh with step size ∆ tmin = N − where 1 /υ > 1. O The error compared to the analytical solution is measured only in time as the spatial behavior of the solution is constant due to the geometry and loading. As an error measure, we employ a discrete L2-error by integrating numerically over the time with a trapzoidal rule. This results in

N err = ∆t (u (t ) u (t )) 2 abs v n n − h n un=0 uX (23) t 1 N − 2 2 err rel = err abs ∆tn (u (tn)) , #n=0 ! X evaluated at some arbitrary point on the surface for the density and at x = ⊺ (1 .5, 0, 0) for the pressure ﬁeld. The index () h indicates the boundary element

19 solution, whereas the other quantity is the respective exact solution. The order of the numerical convergence (eoc) is deﬁned with err eoc = log h , (24) 2 err h+1 where the indices () h+1 and () h denotes two subsequent reﬁnement levels.

170 For the following tests four diﬀerent meshes of the sphere are used (see Fig. 1). The triangles are uniformly reﬁned by subdividing from mesh 1 to mesh 4. The corresponding data as element, node numbers, and respective

(a) mesh 1 (b) mesh 2 (c) mesh 3

(d) mesh 4

Figure 1: Spatial discretisations for the sphere

characteristic element size h are given in Tab. 1. Note that the values of ∆ tconst and h are rounded while, for the computations, more digits are used. Mesh 1

175 consists of 16 elements on a great circle of the sphere. Based on this number the

values of ∆ tconst are computed. To characterize the relation between the spatial

and temporal mesh the dimensionless parameter β = c∆tconst /h is introduced and also given in Tab. 1. The chosen data result in an observation period up

20 Number h β = 0 .25 β = 0 .125 β = 0 .0625

elements nodes ∆tconst N ∆tconst N ∆tconst N mesh 1 128 66 0.393 m 0.000286 11 0.000143 21 0.000072 41 mesh 2 512 258 0.196 m 0.000143 21 0.000072 41 0.000036 82 mesh 3 2048 1026 0.098 m 0.000072 41 0.000036 82 0.000018 163 mesh 4 8192 4098 0.049 m 0.000036 82 0.000018 163 0.000009 270

Table 1: Data of the reﬁned meshes ( β = c∆tconst /h)

to T = 0 .002915905 s, which is approximately the inverse of the wave speed. It

180 must be remarked that for mesh 4 the smallest time step size would have required N = 326 but that many time steps can not be computed on the available hardware. In the following tests, the value υ = 1 /2 is set for the bump function, which ﬁts to the grading parameter χ = 2. In this setting the smallest step size is approximately the square of the largest step size.

185 The first study keeps the value of β constant, hence a refinement in space yields a refinement in the temporal variable as well. The admittance is set to α = 0 .5. The relative error of the density function is plotted in Fig. 2 versus the spatial discretization. To have a comparison, the expected eoc = 1 is given with a dash-dotted line and eoc = 0 .5 with a dotted line. The results show that

190 the method converges with the expected rates. Obviously, the constant time step produces a smaller error but has a smaller convergence order compared to the graded time mesh. Since the pressure is more regular as the density (as explained in the beginning of this section), a better behavior for constant step sizes can be expected. This is somehow conﬁrmed by the results for the

195 pressure. These errors are given in Fig. 3 also versus the spatial discretisation. These solutions of the wave equation are ﬁtting well the expected eoc = 1 also for the constant time mesh and both time meshes behave similar. Apparently,

there is one exceptional point for ∆ tvar , β = 0 .0625. The value for the ﬁnest spatial mesh increases slightly. This might be explained by the non-consistent

200 choices of parameters due to hardware limitation. As already stated, for mesh

21 10 −1.2 ∆tvar , β = 0 .125 ∆ = 0 125 10 −1.4 tconst , β . ∆tvar , β = 0 .0625 −1.6 ∆tconst , β = 0 .0625 10 eoc = 0.5 eoc = 1 10 −1.8

−2

rel 10 err 10 −2.2

10 −2.4

10 −2.6

10 −2.8

10 −1.2 10 −1 10 −0.8 10 −0.6 10 −0.4 mesh size h

Figure 2: Relative error of the density for diﬀerent meshes (double logarithmic scale)

22 10 −1 rel err

∆tvar , β = 0 .25 ∆tconst , β = 0 .25 ∆tvar , β = 0 .125 ∆tconst , β = 0 .125 −2 10 ∆tvar , β = 0 .0625 ∆tconst , β = 0 .0625 eoc = 1 10 −1.2 10 −1 10 −0.8 10 −0.6 10 −0.4 mesh size h

Figure 3: Relative error of the pressure for diﬀerent meshes (double logarithmic scale)

23 4 with β = 0 .0625 the number of time steps to arrive at the ﬁnal time T is too large for the computer memory since all matrices have to be stored for the gCQ method - in contrast to the original CQ method. We expect that this bottleneck can be cured by developing fast versions of gCQ as a topic of future research.

205 Hence, we have ﬁxed the number of time steps for gCQ to N = 270 while, for CQ, we have chosen N = 326. It seems that this shorter observation period deteriorates the error. It must be remarked that the error of the density does not show this behavior. Another view on the convergence behavior is to keep the spatial discretiza-

210 tion constant and reﬁne only in time. Such a test setting may be not recommended in space-time methods as the spatial and temporal behavior of the results are coupled. But in the special setting of this example the spatial result is a constant value and, hence, independent on the spatial mesh. This can also be observed in the results as they are independent on the chosen spatial point

215 on the surface. Therefore, it is possible to study only the quality of the temporal discretization. For this test the time grading is set again to χ = 2. In Fig. 4, the relative error of the density is given versus the time step size for diﬀerent meshes. In addition two eocs are plotted with dashed or dotted lines. They show the expected convergence rates in time as discussed in the beginning of

220 this section. The linear convergence of the graded mesh drops to eoc = 0.5 if a constant time mesh is used. This eﬀect shows that the graded time mesh is able to resolve the non-smooth behavior of the solution induced by the right hand side. The error for the pressure solution shows a diﬀerent behavior. Due to the

225 increased smoothness of the solution also a higher convergence order is expected for the constant time mesh. The results are presented in Fig. 5. The error plots for the constant time mesh show a nearly linear order, whereas the error for the graded time mesh shows partly a quadratic order. A super convergence behavior can be expected from the theory of Galerkin boundary element methods, which

230 might be the reason for this high order. Again, the last point for mesh 4 gets out of the line. As already discussed, this might be an artefact of the changed

24 t 10 −1.2 mesh 2, ∆ const mesh 2, ∆ tvar − . t 10 1 4 mesh 3, ∆ const mesh 3, ∆ tvar 10 −1.6 mesh 4, ∆ tconst mesh 4, ∆ tvar 10 −1.8 eoc = 0.5 eoc = 1

rel 10 −2 err 10 −2.2

10 −2.4

10 −2.6

10 −2.8

10 −5 10 −4 time step size ∆ t

Figure 4: Relative error of the density versus time step size for the diﬀerent meshes (double logarithmic scale)

25 10 −1 rel err mesh 2, ∆ tconst mesh 2, ∆ tvar mesh 3, ∆ tconst mesh 3, ∆ tvar t −2 mesh 4, ∆ const 10 mesh 4, ∆ tvar eoc = 2 eoc = 1

10 −5 10 −4 time step size ∆ t

Figure 5: Relative error of the pressure versus time step size for the diﬀerent meshes (double logarithmic scale)

26 . 0 7 α = 0 α = 0 .25 0.6 α = 0 .5 α = 1 0.5 analytic α = 0 .25 0.4 analytic α = 0 .5 analytic α = 1

ϕ 0.3

0.2 density 0.1

−0.1

−0.2

0 0.2 0.4 0.6 0.8 1 1.2 time t [s] ·10 −2

Figure 6: Density versus time for diﬀerent admittances α (mesh 3)

observation time. Overall, the error plots indicate the improved behavior of the graded time mesh compared to a constant time mesh. After studying the convergence behavior of gCQ, the inﬂuence of the admit-

235 tance α is considered. In Fig. 6, the density is plotted versus time for diﬀerent values of α. We employed mesh 3 for this study and have increased the observation time to T = 0 .0125223 s. Our grading of the time mesh might be sub-optimal for the larger ﬁnal time for the computation of the density since the induced “bump” is periodically repeated in the density (cf. [16]) and, in

240 principle, would require a mesh grading also at later times. However, the inten- tion here is to show the qualitative inﬂuence of the admittance and this might justify a sub-optimal time grading.

27 As expected, the increase of the admittance results in a damping of the solution and a slower decay, where the overall qualitative behavior is similar.

245 Besides the Galerkin boundary element solution with gCQ, the analytical solution is displayed with dashed lines. Only for α = 0 the analytical solution is omitted as it has to be derived separately. It can be observed that the numerical solution cannot follow the peaks after the ﬁrst “bump”. This might be improved by using a grading of the time mesh around these times and indicates

250 that an adaptive time mesh would be advantageous. Further, for larger times the oﬀset to the analytical solution increases. Here the limitations of a BDF 1 as underlying time discretization becomes visible. It can be expected that with a higher order method the approximation of the density becomes better and the generalization to Runge-Kutta gCQ (cf. [19]) is the topic of future research.

255 Interestingly, the pressure solution is not that much inﬂuenced by this devi- ation of the numerical solution. In Fig. 7, the pressure is plotted versus time for the same values of α and the dots display the values of the respective analytical solution. The results match nearly perfect and the inﬂuence of the admittance is the same, it damps the solution. Theoretically this observation can be explained

260 by the smoother behavior of the pressure compared to the density and by well known superconvergence properties of Galerkin BEM for field point evaluations. These numerical experiments confirm that the theoretical findings in Example 4.2 are sharp.

5.2. Improved acoustics in an atrium

265 As a realistic example the influence of absorbing layers in room acoustics is studied. In 2010/11, the atrium of the “Institut für Mathematik” at the University Zurich has been acoustically improved by installing absorber panels at the ceilings. This action has been successful and the following numerical model tries to model this effect.

270 The building is a cube where the oﬃces are located in a ring around the atrium. In Fig. 8, a photo is shown from inside the atrium. On the down side of these ﬂoors sound absorbing material has been mounted. Clearly, the model

28 0.18

0.16

0.14 ] 2 0.12 m / N [ 0.1 u 0.08 α = 0 pressure 0.06 α = 0 .25 0.04 α = 0 .5 α = 1 0.02

0 0.2 0.4 0.6 0.8 1 1.2 time t [s] ·10 −2

Figure 7: Pressure versus time for diﬀerent admittances α (mesh 3)

29 Figure 8: The atrium of the “Institut f¨ur Mathematik” at the University Zurich and the boundary element mesh. The mesh is cut such that the ﬂoors and the stair in the basement are visible.

30 does not include every detail of the geometry, e.g., the construction of the glas roof has not been modelled. However, the geometric model is ﬁne enough, to

275 model details such as the stair from the ground floor to the basement. This simplification allows to compute the sound pressure field in time domain at one node of our cluster (X4800 with 8 OctaCore-Intel CPU & 256GB RAM). This application shows that the proposed BE formulation is capable to treat real world problems.

The material data from air resulting in a wave speed c = 343 m/s are assumed. Further, the time grading is diﬃcult to be adjusted. In contrast to the test before the solution behavior is not known in advance and would in principle require to have an adaptive algorithm. As this is subject to further research, here, a slight modiﬁcation of the grading (22) is used. A smooth increase of time steps sizes is formulated by (n 1) 2 tn = n + − ∆tconst , (25) N !

280 which might be justiﬁed because after several reﬂections in this complicated

geometry the solution behavior does no longer change drastically. ∆ tconst =

0.00037 s is used to discretize the time interval [0 , T = 0 .15 s] and N = T/ ∆tconst = 405 holds. The time grading of (25) results then in 248 time steps to be computed. The chosen constant time step size corresponds to β 0.25. Linear ≈ 4 285 continuous shape functions are employed on 7100 ﬂat triangles . The loading is a given ﬂux at the bottom of the stairs with a time history f (t) = sin (1200 t) H (t) H t 2π . This represents a sine load with 191 Hz · − − 1200 ≈ active over one# period. The# chosen frequency represents a mean frequency of a speaking person.

290 In Fig. 9, the sound pressure in the atrium displayed on a screen placed nearly in the middle of the atrium is depicted for t 0.028 s and t 0.064 s ≈ ≈ for three diﬀerent materials. All walls are assumed to be nearly sound hard

4The geometry and the mesh have been generated by Dominik P¨olz (Graz University of Technology) during his master thesis at the Institute of Applied Mechanics.

31 t ≈ 0.028 s

t ≈ 0.064 s

α = 0 .1 α = 0 .5 α = 1

Figure 9: Sound pressure field in the atrium at different times for three different values of α

32 (e.g., made of concrete with α = 0 .1) but the down side of the ﬂoors, i.e., the ceilings visible in Fig. 8 are modelled as absorbing surfaces. The chosen α-values

295 correspond either to no sound absorbing material ( α = 0 .1), to a heavy curtain at low frequencies ( α = 0 .5), and to the extreme case of totally absorbing surface (α = 1). Note that these values are only examples and may not be the exact values for a distinct material nor the material used in reality in this atrium. The results show clearly traveling waves in the beginning of the computation

300 and a lot of reflections in the complicated geometry. For longer times the sound pressure level approaches a steady state with smaller values for higher damping, i.e., higher admittance. At the beginning, not much differences are visible for different values of the admittance. However, for larger observation times the sound pressure level increases as indicated by the more strong red color for the

305 low damping material compared to the higher damping material. In Fig. 10, the sound pressure level in the upper left side of the screen is exemplarily depicted over time. Note, the sound pressure level is given in dB and negative values indicate sound below the threshold of hearing. Further, the initial phase where the pressure is zero, i.e., the time until the wave arrives, is truncated as in

310 this case the dB measure gives very large negative values. In this plot two things can be observed. First, the peaks with the negative values show the wave reflections, which arrive for the different damping cases at different times. Second, the sound pressure for α = 0 .1, i.e. no mounted damping material, has in the mean the larger pressure values. The two other cases show that the

315 damping material can reduce the sound pressure level as reported from the real building, where it is claimed that the atrium is no longer such noisy. Certainly, the effect is different at different locations within the atrium.

6. Conclusions

Impedance boundary conditions are very natural in acoustic calculations.

320 They constitute a Robin type boundary condition, which can be treated easily with retarded potential integral equations. However, for impedance boundary

33 30 α = 0 .1 α = 0 .5 α 20 = 1 [dB] u 10

−10 sound pressure level

−20

−30 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 time t [s]

Figure 10: Sound pressure level in dB versus time in the upper left part of the observation screen

34 conditions the time derivative of the Dirichlet part is combined with the Neu- mann part. The application of the convolution quadrature (CQ) method as time discretization for this problem is straightforward. Here, the generalized

325 form of the CQ the gCQ is applied, which allows for a non-uniform time mesh in contrast to the original CQ method. For a spherical scatterer, analytical solutions are derived, which are given explicitly for the scattering problem and zero order Hankel functions as spatial right-hand side. In general, the time behavior can be expressed via a time in-

330 tegral while, for a “bump-function”, explicit expressions are presented. These solutions are used to study systematically the convergence behavior of the proposed algorithm with respect to the time discretization. The results show the expected rates. For non-smooth time behavior the gCQ is superior to a constant time mesh as expected. An open question at this point is the optimal grading of

335 the time mesh for more general right-hand sides. Future research will elaborate on adaptivity in time, while the gCQ allows for a straightforward algorithmic realization. Finally, a real world example has been studied. The interior sound ﬁeld in the atrium of the “Institut f¨ur Mathematik”at the University Zurich has been

340 calculated for diﬀerent sound absorbing materials at the ceilings. The calculations show that the proposed method is suitable to compute such a real world application. Certainly, so-called fast methods can improve the performance of the presented formulation.

Acknowledgement. The second author gratefully acknowledges the hospitality

345 and support by the “Institut f¨ur Mathematik”at the University Zurich during his sabbatical leave.

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