The panel-clustering method for the wave equation in two spatial dimensions

Abstract

We consider the numerical solution of the wave equation in a two-dimensional domain and start from a boundary integral formulation for its discretization. We employ the convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. Our main focus is the sparse approximation of the arising sequence of boundary integral operators by panel clustering. This requires the definition of an appropriate admissibility condition such that the arising kernel functions can be efficiently approximated on admissible blocks. The resulting method has a complexity of $\mathcal{O}(\mathit{N}(\mathit{N}+\mathit{M})\mathit{q}^{4+\mathit{s}})$, $\mathit{s} \in$ {0,1}, where $\mathit{N}$ is the number of time points, $\mathit{M}$ denotes the dimension of the boundary element space, and $\mathit{q}=\mathcal{O}$ (log $\mathit{N}$ + log $\mathit{M})$ is the order of the panel-clustering expansion. Numerical experiments will illustrate the efficiency and accuracy of the proposed CQ-BEM method with panel clustering.