Averaging techniques for time-marching schemes for retarded potential integral equations

Abstract Numerical schemes for time domain retarded potential integral equations (RPIEs) are often found to be unstable, having errors which oscillate and grow exponentially with time. It is well known that averaging the solution in time can make such schemes stable by filtering out the high frequency components. However this has to be done carefully—we consider two similar averaging formulae and show that although one stabilizes the underlying scheme, the other actually destabilizes it by amplifying low spatial frequency modes. We carry out a Fourier stability analysis for each of these schemes and show that averaging the solution in space can also successfully remove unstable high spatial frequency modes. Finally we discuss the effect of the spatial approximation and quadrature formula used to approximate the integral by considering a “space-exact” model problem, and present numerical results for a scalar RPIE on both flat and curved surfaces.

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