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The $l^1$ global decay to discrete shocks for scalar monotone schemes


Author: Hailiang Liu
Journal: Math. Comp. 72 (2003), 227-245
MSC (2000): Primary 35L65, 65M06, 65M15
DOI: https://doi.org/10.1090/S0025-5718-01-01380-1
Published electronically: September 17, 2001
MathSciNet review: 1933819
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a family of discrete shocks $\phi$ of a monotone scheme, we prove that the discrete shock profile with rational shock speed $\eta$ is asymptotically stable with respect to the $l^1$ topology $\Vert\cdot\Vert _1$: if $u^0-\phi \in l^1$, then $\Vert u^n -\phi_{\cdot- n\eta}\Vert _1 \to 0$ as $ n\to \infty$ under no restriction conditions of the initial data to the interval $[\inf \phi, \sup \phi]$. The asymptotic wave profile is uniquely identified from the above family by a mass parameter.


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Additional Information

Hailiang Liu
Affiliation: UCLA, Mathematics Department, Los Angeles, California 90095-1555
Email: hliu@math.ucla.edu

DOI: https://doi.org/10.1090/S0025-5718-01-01380-1
Keywords: $l^1$ decay, discrete shocks, monotone scheme
Received by editor(s): December 13, 1999
Received by editor(s) in revised form: November 16, 2000, and January 3, 2001
Published electronically: September 17, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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