Rapid Solution of the Wave Equation in Unbounded Domains: Abridged Version

Abstract We propose and analyze a new fast method for the numerical solution of time-domain boundary integral formulations of the wave equation. Discretization in time is achieved by Lubich’s convolution quadrature method and in space by a Galerkin boundary element method. We show that the arising block Toeplitz system is after a small perturbation equivalent to a a decoupled system of discretized Helmholtz equations. Each of these systems can efficiently be solved by a fast data-sparse method (e.g. FMM, panel clustering). Further savings can be achieved by noticing that in some cases the solutions of many of the Helmholtz problems can be replaced by zero. Finally the proposed method is inherently parallel. We prove that the excellent stability and optimal convergence of the convolution quadrature are inherited by the new method. These results thereby pave the way to the efficient solution using fast data-sparse techniques.