Skip to main content

Difference schemes for nonlinear hyperbolic systems — A general framework

Numerical Analysis

Part of the Lecture Notes in Mathematics book series (LNM,volume 1402)

Abstract

For a hyperbolic system of conservation laws, the general form of conservative difference schemes involving two time-levels in an explicit or implicit way is obtained under natural assumptions. General results are shown on the schemes and this framework is used to study implicit schemes of second-order accuracy.

Keywords

  • Hyperbolic System
  • Implicit Scheme
  • Explicit Scheme
  • Usual Scheme
  • Numerical Flux

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. LAX P.D. and WENDROFF B.-Systems of conservation laws, Comm. Pure Appl. Math., 13, pp. 217–237, 1960.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. HARTEN A.-On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. Numer. Anal., 21, pp. 1–23, 1984.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. TADMOR E.-Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comput., 43, pp. 369–381, 1984.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. LAX P.D.-Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math., 7, pp. 159–193, 1954.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. ROE P.L.-Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, pp. 357–372, 1981.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. HARTEN A-On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49, pp. 151–164, 1983.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. LERAT A. and PEYRET R.-Sur le choix de schémas aux différences du second ordre fournissant des profils de choc sans oscillation, C.R. Acad. Sci. Paris, 277 A, pp. 363–366, 1973.

    MathSciNet  MATH  Google Scholar 

  8. RICHTMYER R.D. and MORTON K.W.-Difference methods for initial-value problems, Interscience Publ., New York, 1967.

    MATH  Google Scholar 

  9. MacCORMACK R.W.-The effect of viscosity in hypervelocity impact cratering, AIAA Paper no 69–354, 1969.

    Google Scholar 

  10. BEAM R. and WARMING R.F.-An implicit finite-difference algorithm for hyperbolic systems in conservation-law form, J. Comput. Phys., 22, pp. 87–110, 1976.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. LERAT A.-Une classe de schémas aux différences implicites pour les systèmes hyperboliques de lois de conservation, C.R. Acad. Sci. Paris, 288 A, pp. 1033–1036, 1979.

    MathSciNet  MATH  Google Scholar 

  12. DARU V. and LERAT A.-Analysis of an implicit Euler solver, in Numerical Methods for the Euler Equations of Fluid Dynamics, F. Angrand et al. Eds, SIAM Publ., pp. 246–280, 1985.

    Google Scholar 

  13. LERAT A. and SIDES J.-Efficient solution of the steady Euler equations with a centered implicit method, Intern. Conf. Num. Meth. Fluid Dyn., Oxford, march 1988, To appear. Also TP ONERA 1988-128.

    Google Scholar 

  14. OSHER S.-Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal., 21, pp. 217–235, 1984.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. TADMOR E.-The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comput., 49, pp. 91–103, 1987.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. KHALFALLAH K. and LERAT A.-Correction d'entropie pour des schémas numériques approchant un système hyperbolique, to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Lerat, A. (1989). Difference schemes for nonlinear hyperbolic systems — A general framework. In: Carasso, C., Charrier, P., Hanouzet, B., Joly, JL. (eds) Nonlinear Hyperbolic Problems. Lecture Notes in Mathematics, vol 1402. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083865

Download citation

  • DOI: https://doi.org/10.1007/BFb0083865

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51746-7

  • Online ISBN: 978-3-540-46800-4

  • eBook Packages: Springer Book Archive