Nonlinear stability of discrete shocks for systems of conservation laws

Jian-Guo Liu Courant Institute of Mathematical Sciences Zhouping Xin Courant Institute of Mathematical Sciences

Analysis of PDEs mathscidoc:1702.03076

Archive for Rational Mechanics and Analysis, 125, (3), 217–256, 1993.9
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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@inproceedings{jian-guo1993nonlinear,
  title={Nonlinear stability of discrete shocks for systems of conservation laws},
  author={Jian-Guo Liu, and Zhouping Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209142248833907412},
  booktitle={Archive for Rational Mechanics and Analysis},
  volume={125},
  number={3},
  pages={217–256},
  year={1993},
}
Jian-Guo Liu, and Zhouping Xin. Nonlinear stability of discrete shocks for systems of conservation laws. 1993. Vol. 125. In Archive for Rational Mechanics and Analysis. pp.217–256. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209142248833907412.
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