Convolution quadrature and discretized operational calculus I.
Numerische Mathematik
Fast Convolution for Nonreflecting Boundary Conditions
SIAM Journal on Scientific Computing
On the numerical inversion of the Laplace transform of certain holomorphic mappings
Applied Numerical Mathematics
Fast and Oblivious Convolution Quadrature
SIAM Journal on Scientific Computing
Adaptive, Fast, and Oblivious Convolution in Evolution Equations with Memory
SIAM Journal on Scientific Computing
Rapid Solution of the Wave Equation in Unbounded Domains
SIAM Journal on Numerical Analysis
Multistep and Multistage Convolution Quadrature for the Wave Equation: Algorithms and Experiments
SIAM Journal on Scientific Computing
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Linear 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 Schadle 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.