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Convergence rates
to the discrete travelling wave
for relaxation schemes

Author: Hailiang Liu
Journal: Math. Comp. 69 (2000), 583-608
MSC (1991): Primary 35L65, 65M06, 65M12
Published electronically: March 11, 1999
MathSciNet review: 1653958
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Abstract: This paper is concerned with the asymptotic convergence of numerical solutions toward discrete travelling waves for a class of relaxation numerical schemes, approximating the scalar conservation law. It is shown that if the initial perturbations possess some algebraic decay in space, then the numerical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time, provided the sums of the initial perturbations for the $u$-component equal zero. A polynomially weighted $l^2$ norm on the perturbation of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate.

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Additional Information

Hailiang Liu
Affiliation: Department of Mathematics, Henan Normal University, Xinxiang, 453002, China
Address at time of publication: Institute of Analysis and Numerics, Otto-von-Guericke-University Magdeburg, PSF 4120, 39106 Magdeburg, Germany

Keywords: Relaxation scheme, nonlinear stability, discrete travelling wave, convergence rate
Received by editor(s): December 16, 1997
Received by editor(s) in revised form: July 14, 1998
Published electronically: March 11, 1999
Additional Notes: Research supported in part by an Alexander von Humboldt Fellowship at the Otto-von-Guericke-Universität Magdeburg, and by the National Natural Science Foundation of China
Article copyright: © Copyright 2000 American Mathematical Society

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