Archive for Rational Mechanics and Analysis

, Volume 125, Issue 3, pp 217–256 | Cite as

Nonlinear stability of discrete shocks for systems of conservation laws

  • Jian -Guo Liu
  • Zhouping Xin
Article

Abstract

In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the Lp-norm for all p ≧ 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in Lp (P ≧ 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.

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References

  1. 1.
    I-Ling Chern, Large-time behavior of solutions of Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws, Math. Comp. 56 (1991), 107–118.Google Scholar
  2. 2.
    M. G. Crandall & A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 1–21.Google Scholar
  3. 3.
    R. Diperna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27–70.Google Scholar
  4. 4.
    B. Engquist & S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 35 (1981), 321–351.Google Scholar
  5. 5.
    J. Goodman, Nonlinear asymptotic stability of viscous shock profiles to conservation laws, Arch. Rational Mech. Anal. 95 (1986), 325–344.CrossRefGoogle Scholar
  6. 6.
    J. Goodman, Remarks on the stability of viscous shock waves, Viscous Profiles and Numerical Methods for Shock Waves, Edited by M. Shearer, SIAM, Philadelphia, 1991.Google Scholar
  7. 7.
    J. Goodman & Zhouping Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal. 121 (1992), 235–265.CrossRefGoogle Scholar
  8. 8.
    E. Harabetian, Rarefaction and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), 527–536.Google Scholar
  9. 9.
    G. H. Hardy, J. E. Littlewood & G. Polya, Inequalities, Cambridge University Press, London, New York, 1943.Google Scholar
  10. 10.
    G. Jennings, Discrete shocks, Comm. Pure Appl. Math. 27 (1974), 25–37.Google Scholar
  11. 11.
    S. Kawashima & A. Matsumura, Asymptotic stability of traveling wave solution to system for one dimensional gas motion, Comm. Math. Phys. 101 (1985), 97–127.Google Scholar
  12. 12.
    N. N. Kuznecov & S. A. Vološin, On monotone difference approximations of a first order quasilinear equation, Soviet Math. Dokl. 17 (1976), 1203–1206.Google Scholar
  13. 13.
    P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954), 159–193.Google Scholar
  14. 14.
    P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973.Google Scholar
  15. 15.
    Tai-Ping Liu, Nonlinear Stability of Shock Waves for Viscous Conservation Laws, Mem. Amer. Math. Soc. 328 Amer. Math. Soc., Providence, 1985.Google Scholar
  16. 16.
    Jian-Guo Liu & Zhouping Xin, L 1-stability of stationary discrete shocks, Math. Comp. 60 (1993), 233–244.Google Scholar
  17. 17.
    Jian-Guo Liu & Zhouping Xin, Convergence of Lax-Friedrichs scheme for piecewise smooth solutions to systems of conservation laws, in preparation.Google Scholar
  18. 18.
    A. Majda & J. Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math. 32 (1979), 445–483.Google Scholar
  19. 19.
    D. Michelson, Discrete shocks for difference approximations to systems of conservation laws, Adv. Appl. Math. 5 (1984), 433–469.Google Scholar
  20. 20.
    T. Nishida & J. Smoller, A class of convergent finite-difference schemes for certain nonlinear parabolic systems, Comm. Pure Appl. Math. 36 (1983), 785–808.Google Scholar
  21. 21.
    O. Oleinik, Discontinuous solutions of nonlinear differential equations, Usp. Mat. Nauk. 12 (1957), 3–73.Google Scholar
  22. 22.
    S. Osher & J. Ralston, L 1 stability of traveling waves with applications to convective porous media flow, Comm. Pure Appl. Math. 35 (1982), 737–751.Google Scholar
  23. 23.
    R. D. Richtmyer & K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Wiley-Interscience, New York, 1967.Google Scholar
  24. 24.
    R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91–106.Google Scholar
  25. 25.
    Y. S. Smyrlis, Existence and stability of stationary profiles of the LW scheme, Comm. Pure Appl. Math. 42 (1990), 509–545.Google Scholar
  26. 26.
    A. Szepessy, On the stability of finite element methods for shock waves, Comm. Pure Appl. Math. 45 (1992), 923–946.Google Scholar
  27. 27.
    A. Szepessy & Zhouping Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993), 53–103.Google Scholar
  28. 28.
    E. Tadmor, The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme, Math. Comp. 43 (1984), 353–368.Google Scholar
  29. 29.
    Zhouping Xin, On the linearized stability of viscous shock profiles for systems of conservation laws, J. Diff. Eq. 100 (1992), 119–136.Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Jian -Guo Liu
    • 1
  • Zhouping Xin
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York

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