Nonlinear stability of discrete shocks for systems of conservation laws
 Jian Guo Liu,
 Zhouping Xin
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In this paper we study the asymptotic nonlinear stability of discrete shocks for the LaxFriedrichs 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 LaxFriedrichs scheme for the singleshock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given farfield 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 LaxFriedrichs scheme.
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 Title
 Nonlinear stability of discrete shocks for systems of conservation laws
 Journal

Archive for Rational Mechanics and Analysis
Volume 125, Issue 3 , pp 217256
 Cover Date
 19930901
 DOI
 10.1007/BF00383220
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 Jian Guo Liu ^{(1)}
 Zhouping Xin ^{(1)}
 Author Affiliations

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