On superconvergence of Runge-Kutta convolution quadrature for the wave equation
Jens Markus Melenk, Alexander Rieder
TL;DR
The paper analyzes time-discretized boundary-integral formulations for the wave equation in sound-soft scattering, using Runge-Kutta convolution quadrature (RK-CQ). It explains how a differentiated-data formulation yields higher convergence orders by leveraging a decomposition $s^{-1}\operatorname{DtN}(s)=\operatorname{DtI}(s)+I$ and a refined bound for the Dirichlet-to-Impedance map, which is sharper than the Dirichlet-to-Neumann bound on smooth and polygonal geometries (with an extra log term for polygons). The authors establish a standard RK-CQ convergence rate and prove an improved rate for differentiated data, supported by a detailed analysis of DtN/DtI in both smooth and polygonal domains, including boundary- and corner-layer constructions. Numerically, the results are shown to be sharp, and the work provides a rigorous explanation for observed superconvergence phenomena in wave scattering problems, with guidance on choosing problem formulations and regularity assumptions in practice.
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.