MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

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

Lehel Banjai and Volker Gruhne


Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be rewritten as a time-domain boundary integral equation. 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 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. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.

Oct 29, 2010
Nov 2, 2010
MSC Codes:
65M38, 35L05
boundary element method, convolution quadrature, wave equation

Related publications

2011 Repository Open Access
Lehel Banjai and Volker Gruhne

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

In: Journal of computational and applied mathematics, 235 (2011) 14, pp. 4207-4220