Richardson Extrapolation for Linearly Degenerate Discontinuities

Authors:
J. W. Banks;T. D. Aslam
Affiliations:
Center for Applied Scientific Computing, L-422, Lawrence Livermore National Laboratory, Livermore, USA 94551;Weapons Experiments, MS P952, Los Alamos National Laboratory, Los Alamos, USA 87545
Venue:
Journal of Scientific Computing
Year:
2013

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Abstract

In this paper we investigate the use of Richardson extrapolation to estimate the convergence rates for numerical solutions to wave propagation problems involving discontinuities. For many cases, we find that the computed results do not agree with the a-priori estimate of the convergence rate. Furthermore, the estimated convergence rate is found to depend on the specific details of how Richardson extrapolation was applied; in particular the order of comparisons between three approximate solutions can have a significant impact. Modified equations are used to analyze the situation. We elucidated, for the first time, the cause of apparently unpredictable estimated convergence rates from Richardson extrapolation in the presence of discontinuities. Furthermore, we ascertain one correct structure of Richardson extrapolation that can be used to obtain predictable estimates. We demonstrate these results using a number of numerical examples.