Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

Abstract

a b s t r a c tLinear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be reformulated as time-domain boundary integral equations. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens' principle does not hold.For the numerical discretization, we make use of convolution quadrature in time and standard Galerkin boundary element method in space. The quadrature in time results in a discrete convolution of weights W j with the boundary density evaluated at equally spaced time points. If the strong Huygens' principle holds, W j converge to 0 exponentially quickly for large enough j. If the strong Huygens' principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation.In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. Further, we hint on the possibility to apply the fast and oblivious convolution quadrature algorithm of Schädle et al. to further reduce memory requirements for long-time computation. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.

Some properties of layer potentials and boundary integral operators for the wave equation

“…Not surprisingly, when Lubich's convolution quadrature techniques started to be applied to retarded boundary integral equations (this happened in [13]), the key results of Bamberger and Ha-Duong were instrumental in proving convergence estimates for a method that relies heavily on the Laplace transform of the symbol of the operator, even though it is a marching-on-in-time scheme. The relevance of having precise bounds in the Laplace domain for numerical analysis purposes has also been expanded in more recent work at the abstract level (with the recent analysis of RK-CQ schemes in [5] and [6]) and with applications to the wave equation at different stages of discretization ( [12], [4], [8])…”

Fast and Oblivious Algorithms for Dissipative and Two-dimensional Wave Equations

“…in three dimensions, (5.4) where K 0 (·) is a modified Bessel function. The transfer function satisfies the bounds (2.2) and (2.1) as first noticed in [5]. Lemma 9.…”

“…where K 0 (·) is a modified Bessel function [1]. This function satisfies the bounds (2.2) and (2.1) with µ = −1/2 as proved in [5]. In Figures 2 and 3 we show the error in the approximation of the convolution weights in (4.10) along approximation intervals of the form [hn 0 + hB ℓ , hn 0 + 2hB ℓ+1 ], with B = 5 and B = 10 and for two different values of the distance parameter d, namely d = 0.1 (Fig.…”

“…The smoothness of this tail has been used in [10] and [5] to speed up computations and more importantly for the current work, in [5] it was noticed that this tail is due to an operator of parabolic type; the precise meaning of this is explained in the next section. This fact was used in [5] for purely theoretical purposes whereas in this paper it will be used to develop a fast algorithm with reduced memory requirements. Our new algorithm applied to the tail of the linear hyperbolic problems has essentially the same properties as the algorithms developed in [25,30] for parabolic problems.…”

Functional estimates for derivatives of the modified Bessel functionK0and related exponential functions

“…As is common in applied mathematics new questions arise from new applications; in this case, the rapidly increasing interest in convolution quadrature methods for solving the wave equation in exterior domains (cf. [10,11,13,12,9,2]) leads to the question of functional-type inequalities for derivatives of the Bessel functions which are explicit with respect to the order of the derivatives. This has been invested for the three-dimensional case in [8], where the kernel function is the exponential function.…”