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.
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This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was funded by the Uncertainty Quantification Strategic Initiative Laboratory Directed Research and Development Project at LLNL under project tracking code 10-SI-013, by DOE contracts from the ASCR Applied Math Program, and by Los Alamos National Laboratory under Contract DE-AC52-06NA25396.
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Banks, J.W., Aslam, T.D. Richardson Extrapolation for Linearly Degenerate Discontinuities. J Sci Comput 57, 1–18 (2013). https://doi.org/10.1007/s10915-013-9693-0
- Richardson extrapolation
- Error estimation
- Convergence analysis
- Shock capturing