L. Banjai, C. Lubich, J. Melenk:

"Runge-Kutta convolution quadrature for operators arising in wave propagation";

in: "ASC Report 24/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

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 $\BigO(s^{\mu_1})$ there,

the stronger polynomial bound $\BigO(s^{\mu_2})$ 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.

http://www.asc.tuwien.ac.at/preprint/2010/asc24x2010.pdf

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