Skip to main content

Richardson Extrapolation for Linearly Degenerate Discontinuities

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Giles, M.B., üli, E.S.: Acta Numerica 11, 145 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Estep, D., Larson, M., Williams, R.: Mem. Am. Math. Soc. 696, 1 (2000)

    Google Scholar 

  3. 3.

    Hay, A., Visonneau, M.: Int. J. Comput. Fluid D. 20(7), 463 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Banks, J.W., Hittinger, J.A.F., Connors, J.M., Woodward, C.S.: Comput. Method. Appl. Mech. Engrg. 213—-216, 1 (2012)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ainsworth, M., Oden, J.T.: Comput. Method. Appl. Mech. Engrg. 142, 1 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Roy C.J. Review of discretization error estimators in scientific computing. AIAA Paper 2010–126 (2010).

  7. 7.

    Niethammer, R., Kim, K., Ballmann, J.: Int. J. Impact Eng. 16, 1711 (1995)

    Article  Google Scholar 

  8. 8.

    Appelö, D., Banks, J.W., Henshaw, W.D., Schwendeman, D.W.: J. Comput. Phys. 231, 6012 (2012)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philidelphia. (1972)

    Google Scholar 

  10. 10.

    Banks, J.W., Henshaw, W.D., Schwendeman, D.W., Kapila, A.K.: Combust. Theory and Modelling 12(4), 769 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Banks, J.W., Aslam, T., Rider, W.J.: J. Comput. Phys. 227(14), 6985 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Henshaw, W.D., Schwendeman, D.W.: J. Comput. Phys. 216(2), 744 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Banks, J.W., Henshaw, W.D., Shadid, J.N.: J. Comput. Phys. 228(15), 5349 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Banks, J.W., Henshaw, W.D.: J. Comput. Phys. 231(17), 5854 (2012)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience, New York (1974)

    MATH  Google Scholar 

  16. 16.

    Lax, P., Wendroff, B.: Commun. Pur. Appl. Math. 13, 217 (1960)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Després, B.: Numer. Math. 108, 529 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Després, B.: SIAM J. Numer. Anal. 47, 3956 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    van Leer, B.: J. Comput. Phys. 32, 101 (1979)

    Article  Google Scholar 

  20. 20.

    Sweby, P.K.: SIAM J. Numer. Anal. 21, 995 (1984)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to J. W. Banks.

Additional information

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.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Keywords

  • Richardson extrapolation
  • Error estimation
  • Convergence analysis
  • Shock capturing