Runge–Kutta convolution quadrature based on Gauss methods

Abstract

An error analysis of Runge–Kutta convolution quadrature based on Gauss methods applied to hyperbolic operators is given. Order reduction is observed, with the order of convergence depending heavily on the parity of the number of stages, a more favourable situation arising for the odd cases than the even ones. An exception is observed when the associated kernel exhibits exponential decay. In this case, for the 2-stage Gauss method full order is obtained. For particular situations the order of convergence is higher than for Radau IIA or Lobatto IIIC methods when using the same number of odd stages. We 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 illustrate the performance of the method.