A Lax-Wendroff type theorem for unstructured quasi-uniform grids

Author:
Volker Elling

Journal:
Math. Comp. **76** (2007), 251-272

MSC (2000):
Primary 65M12; Secondary 35L65

DOI:
https://doi.org/10.1090/S0025-5718-06-01881-3

Published electronically:
August 22, 2006

MathSciNet review:
2261020

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Abstract: A well-known theorem of Lax and Wendroff states that if the sequence of approximate solutions to a system of hyperbolic conservation laws generated by a conservative consistent numerical scheme converges boundedly a.e. as the mesh parameter goes to zero, then the limit is a weak solution of the system. Moreover, if the scheme satisfies a discrete entropy inequality as well, the limit is an entropy solution. The original theorem applies to uniform Cartesian grids; this article presents a generalization for quasi-uniform grids (with Lipschitz-boundary cells) uniformly continuous inhomogeneous numerical fluxes and nonlinear inhomogeneous sources. The added generality allows a discussion of novel applications like local time stepping, grids with moving vertices and conservative remapping. A counterexample demonstrates that the theorem is not valid for arbitrary non-quasi-uniform grids.

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Additional Information

**Volker Elling**

Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02906

Email:
velling@stanfordalumni.org

DOI:
https://doi.org/10.1090/S0025-5718-06-01881-3

Keywords:
Finite volume method,
conservation law,
convergence,
Lax--Wendroff,
conservative remapping

Received by editor(s):
April 21, 2003

Received by editor(s) in revised form:
October 20, 2005

Published electronically:
August 22, 2006

Additional Notes:
This material is based upon work supported by an SAP/Stanford Graduate Fellowship and by the National Science Foundation under Grant no. DMS 0104019. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Article copyright:
© Copyright 2006
Volker Elling