Runge–Kutta approximation for $$C_0$$-semigroups in the graph norm with applications to time domain boundary integral equations

Abstract

We consider the approximation of an abstract evolution problem with inhomogeneous side constraint using A-stable Runge–Kutta methods. We derive a priori estimates in norms other than the underlying Banach space. Most notably, we derive estimates in the graph norm of the generator. These results are used to study convolution quadrature based discretizations of a wave scattering and a heat conduction problem.

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

“…When considering discretizations of the wave equation using boundary integral methods, this is not always the case. Instead, it has been observed that sometimes a "superconvergence phenomenon" appears, where the observed convergence rate surpasses those predicted, see [RSM19a,RSM19b,Rie17].…”

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

“…When considering discretizations of the wave equation using boundary integral methods, this is not always the case. Instead, it has been observed that sometimes a "superconvergence phenomenon" appears, where the observed convergence rate surpasses those predicted, see [29][30][31].…”

Time domain boundary integral equations and convolution quadrature for scattering by composite media (extended preprint)

“…In this paper, we present and analyze a fully discrete formulation of the multiplesubdomain acoustic scattering problem based on a Galerkin boundary element method and RK-CQ for the time discretization. Our analysis is based on a pure time-domain point of view, combining ideas by [BLS15a] and [HQSVS17] with the theory of Runge-Kutta approximations of abstract semigroups, as laid out in [AMP03] and recently extended in [RSM20]. A main contribution of the present work is that our analysis covers scattering problems by piecewise constant materials with a very general layout of subdomains.…”

“…Our approach therefore generalizes the results of [Qiu16, Chapters 3 and 4], which only allows certain nested geometries (see Section 5). Compared to other works, e.g., [QS16,Qiu16] we also consider RK-CQ using the novel time-domain analysis developed in [RSM20], whereas previous analyses concentrated on multistep methods, whose order, however, is limited to 2 if A-stability is required. These higher order RK-methods suffer from some reduction of order phenomenon in that the convergence order falls somewhere between the stage-and classical order of the RK-method.…”

“…In contrast, our analysis includes a fully discrete convergence analysis for more complicated geometric situations of arbitrary number of subdomains. In order to do so, we use the novel "time-domain only approach" developed along side this paper and presented recently in [RSM20], showcasing that this approach is feasible for complex model problems that go beyond rather simple ones.…”