Overresolving in the Laplace domain for convolution quadrature methods

TL;DR

This paper explores overresolving in the Laplace domain for convolution quadrature methods, enabling more Laplace domain solutions than time steps, and analyzes how pole locations affect accuracy in wave problem simulations.

Contribution

It introduces a novel approach to decouple Laplace domain solves from time steps, allowing overresolution, and provides error analysis based on complex approximation theory.

Findings

Overresolving improves accuracy in CQ methods.

Pole locations of the solution operator influence error.

Numerical examples confirm theoretical predictions.

Abstract

Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ type formulations for the parallel space-time evaluation of multistep or stiffly accurate Runge-Kutta rules for the wave equation. In particular, we decouple the number of Laplace domain solves from the number of time steps. This allows to overresolve in the Laplace domain by computing more Laplace domain solutions solutions than there are time steps. We use techniques from complex approximation theory to analyse the error of the CQ approximation of the underlying time-stepping rule when overresolving in the Laplace domain and show that the performance is intimately linked to the location of the poles of the solution operator. Several examples using boundary integral equation formulations in the Laplace domain are presented to illustrate the main results.