On superconvergence of Runge-Kutta convolution quadrature for the wave equation

Abstract

The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge-Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of $\abs{s}$(up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map.

References in corpus (4)

• Boundary elements with mesh refinements for the wave equation
• Time domain boundary integral equations and convolution quadrature for scattering by composite media (extended preprint)
• Stable decompositions of $hp$-BEM spaces and an optimal Schwarz preconditioner for the hypersingular integral operator in 3D
• Runge-Kutta approximation for $C_0$-semigroups in the graph norm with applications to time domain boundary integral equations

Cited by in corpus (2)

• Time domain boundary integral equations and convolution quadrature for scattering by composite media (extended preprint)
• Runge-Kutta approximation for $C_0$-semigroups in the graph norm with applications to time domain boundary integral equations