Advertisement

Difference schemes for nonlinear hyperbolic systems — A general framework

  • A. Lerat
Numerical Analysis General Theory
  • 335 Downloads
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    LAX P.D. and WENDROFF B.-Systems of conservation laws, Comm. Pure Appl. Math., 13, pp. 217–237, 1960.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    HARTEN A.-On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. Numer. Anal., 21, pp. 1–23, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    TADMOR E.-Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comput., 43, pp. 369–381, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    LAX P.D.-Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math., 7, pp. 159–193, 1954.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    ROE P.L.-Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, pp. 357–372, 1981.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    HARTEN A-On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49, pp. 151–164, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [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.MathSciNetzbMATHGoogle Scholar
  8. [8]
    RICHTMYER R.D. and MORTON K.W.-Difference methods for initial-value problems, Interscience Publ., New York, 1967.zbMATHGoogle Scholar
  9. [9]
    MacCORMACK R.W.-The effect of viscosity in hypervelocity impact cratering, AIAA Paper no 69–354, 1969.Google Scholar
  10. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [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.MathSciNetzbMATHGoogle Scholar
  12. [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. [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. [14]
    OSHER S.-Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal., 21, pp. 217–235, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    TADMOR E.-The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comput., 49, pp. 91–103, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    KHALFALLAH K. and LERAT A.-Correction d'entropie pour des schémas numériques approchant un système hyperbolique, to appear.Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • A. Lerat
    • 1
    • 2
  1. 1.ENSAMParisFrance
  2. 2.ONERAChâtillonFrance

Personalised recommendations