Runge-Kutta convolution quadrature based on Gauss methods

TL;DR

This paper analyzes the convergence properties of Runge-Kutta convolution quadrature based on Gauss methods for hyperbolic operators, highlighting the influence of stage parity and demonstrating superior convergence in specific applications like acoustic scattering.

Contribution

It provides a detailed error analysis of Gauss-based Runge-Kutta convolution quadrature, revealing improved convergence for odd stages and in certain scattering problems.

Findings

Convergence order depends on the parity of the number of stages.

Odd-stage methods exhibit more favorable convergence properties.

Numerical experiments confirm the theoretical advantages in acoustic scattering applications.

Abstract

An error analysis of Runge-Kutta convolution quadrature based on Gauss methods applied to hyperbolic operators is given. The order of convergence relies heavily on the parity of the number of stages, a more favourable situation arising for the odd cases than the even ones. Moreover, for particular situations the order of convergence is higher than for Radau IIA or Lobatto IIIC methods when using the same number of stages. We further investigate an application to transient acoustic scattering where, for certain scattering obstacles, the favourable situation occurs in the important case of the exterior Dirichlet-to-Neumann map. Numerical experiments and comparisons show the performance of the method.