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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.