Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
59/2010

Runge-Kutta convolution quadrature for operators arising in wave propagation

Lehel Banjai, Christian Lubich and Jens Markus Melenk

Abstract

An error analysis of Runge-Kutta convolution quadrature is presented for a class of non-sectorial operators whose Laplace transform satisfies, besides the standard assumptions of analyticity in a half-plane $\Re s > \sigma_0$ and a polynomial bound $O(s^{\mu_1})$ there, the stronger polynomial bound $O(s^{\mu_2})$ in convex sectors of the form $|\operatorname{arg} s| \leq \pi/2-\theta < \pi/2$ for $\theta > 0$. The order of convergence of the Runge-Kutta convolution quadrature is determined by $\mu_2$ and the underlying Runge-Kutta method, but is independent of $\mu_1$.

Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of Runge-Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical order of the Runge-Kutta method is attained away from the scattering boundary.

Received:
Oct 11, 2010
Published:
Oct 19, 2010
MSC Codes:
65R20, 65L06, 65M15
Keywords:
convolution quadrature, Runge-Kutta methods, Time-domain boundary integral operators, order reduction

Related publications

inJournal
2011 Repository Open Access
Lehel Banjai, Christian Lubich and Jens Markus Melenk

Runge-Kutta convolution quadrature for operators arising in wave propagation

In: Numerische Mathematik, 119 (2011) 1, pp. 1-20