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Existence and asymptotic stability of relaxation discrete shock profiles

Author: Mao Ye
Journal: Math. Comp. 73 (2004), 1261-1296
MSC (2000): Primary 65M12; Secondary 35L65
Published electronically: March 3, 2004
MathSciNet review: 2047087
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Abstract: In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in $L^2$, 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 relaxing scheme for the single shock solution of the system of hyperbolic conservation laws. These results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks.

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Additional Information

Mao Ye
Affiliation: Institute of Mathematics Science and Department of Mathematics, Chinese University of Hong Kong, Hong Kong
Address at time of publication: School of Computer Science and Engineering, University of Electronic Science and Technology of China, Sichuan, China 610054

Keywords: Relaxing scheme, hyperbolic systems of conservation laws, discrete shock profiles, nonlinear stability
Received by editor(s): September 3, 2002
Published electronically: March 3, 2004
Additional Notes: This work was supported by the Youth Science and Technology Foundation, UESTC YF020801.
Article copyright: © Copyright 2004 American Mathematical Society