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Nonlinear stability of discrete shocks for systems of conservation laws

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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 L p-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 L p (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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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 10012, New York, New York

    Jian -Guo Liu & Zhouping Xin

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  1. Jian -Guo Liu
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  2. Zhouping Xin
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Communicated by C. Dafermos

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Liu, J.G., Xin, Z. Nonlinear stability of discrete shocks for systems of conservation laws. Arch. Rational Mech. Anal. 125, 217–256 (1993). https://doi.org/10.1007/BF00383220

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