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Existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws


Author: Haitao Fan
Journal: Math. Comp. 70 (2001), 1043-1069
MSC (2000): Primary 80A32, 35L65, 35L67
DOI: https://doi.org/10.1090/S0025-5718-00-01254-0
Published electronically: May 23, 2000
MathSciNet review: 1826576
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Abstract:

When shock speed $s$ times $\Delta t/\Delta x$ is rational, the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved. A sufficient condition under which the above solutions can be extended to $-\infty <j<\infty $, implying the existence of discrete shock profiles of numerical schemes, is provided. A class of monotonicity preserving schemes, including all monotonicity preserving schemes with $C^{1}$ numerical flux functions, the second order upwinding flux based MUSCL scheme, the second order flux based MUSCL scheme with Lax-Friedrichs' splitting, and the Godunov scheme for scalar conservation laws are found to satisfy this condition. Thus, the existence of discrete shock profiles for these schemes is established when $s\Delta t/\Delta x$ is rational.


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Additional Information

Haitao Fan
Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057
Email: fan@archimedes.math.georgetown.edu

DOI: https://doi.org/10.1090/S0025-5718-00-01254-0
Keywords: Discrete shock profile, discrete traveling wave, monotonicity preserving scheme, MUSCL scheme, conservation laws
Received by editor(s): October 23, 1998
Received by editor(s) in revised form: July 23, 1999
Published electronically: May 23, 2000
Additional Notes: Research supported by NSF Fellowship under Grant DMS-9306064.
Article copyright: © Copyright 2000 American Mathematical Society

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