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

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- by Volker Elling;
- Math. Comp.
**76**(2007), 251-272 - DOI: https://doi.org/10.1090/S0025-5718-06-01881-3
- Published electronically: August 22, 2006

## 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.## References

- Alberto Bressan and Philippe LeFloch,
*Uniqueness of weak solutions to systems of conservation laws*, Arch. Rational Mech. Anal.**140**(1997), no. 4, 301–317. MR**1489317**, DOI 10.1007/s002050050068 - Bernardo Cockburn, Frédéric Coquel, and Philippe LeFloch,
*An error estimate for finite volume methods for multidimensional conservation laws*, Math. Comp.**63**(1994), no. 207, 77–103. MR**1240657**, DOI 10.1090/S0025-5718-1994-1240657-4 - Frédéric Coquel and Philippe LeFloch,
*Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach*, Math. Comp.**57**(1991), no. 195, 169–210. MR**1079010**, DOI 10.1090/S0025-5718-1991-1079010-2 - Frédéric Coquel and Philippe LeFloch,
*Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory*, SIAM J. Numer. Anal.**30**(1993), no. 3, 675–700. MR**1220646**, DOI 10.1137/0730033 - Michael G. Crandall and Andrew Majda,
*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, DOI 10.1090/S0025-5718-1980-0551288-3 - Michael G. Crandall and Luc Tartar,
*Some relations between nonexpansive and order preserving mappings*, Proc. Amer. Math. Soc.**78**(1980), no. 3, 385–390. MR**553381**, DOI 10.1090/S0002-9939-1980-0553381-X - J. K. Dukowicz and J. R. Baumgardner,
*Incremental remapping as a transport/advection algorithm*, J. Comput. Phys.**160**(2000), 318–335. - John K. Dukowicz and John W. Kodis,
*Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations*, SIAM J. Sci. Statist. Comput.**8**(1987), no. 3, 305–321. MR**883773**, DOI 10.1137/0908037 - J. K. Dukowicz,
*Conservative rezoning (remapping) for general quadrilateral meshes*, J. Comput. Phys.**54**(1984), 411–424. - V. Elling,
*A Lax–Wendroff type theorem for semidiscrete schemes on unstructured quasi-uniform grids*, in preparation. - —,
*Methods and theory for conservative remapping*, in preparation. - —,
*Numerical simulation of gas flow in moving domains*, Diploma Thesis, RWTH Aachen (Germany), 2000. - —,
*A possible counterexample to entropy solutions and to Godunov scheme convergence*,**75**(2006), 1721–1733. - Edwige Godlewski and Pierre-Arnaud Raviart,
*Numerical approximation of hyperbolic systems of conservation laws*, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. MR**1410987**, DOI 10.1007/978-1-4612-0713-9 - Jeffrey Grandy,
*Conservative remapping and region overlays by intersecting arbitrary polyhedra*, J. Comput. Phys.**148**(1999), no. 2, 433–466. MR**1669715**, DOI 10.1006/jcph.1998.6125 - A. Harten, J. M. Hyman, and P. D. Lax,
*On finite-difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**(1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR**413526**, DOI 10.1002/cpa.3160290305 - P.W. Jones,
*First- and second-order conservative remapping schemes for grids in spherical coordinates*, Monthly Weather Review (1999), no. 9, 2204–2210. - S. N. Kružkov,
*First order quasilinear equations with several independent variables*, Mat. Sb. (N.S.)**81(123)**(1970), 228–255 (Russian). MR**267257** - D. Kröner, M. Rokyta, and M. Wierse,
*A Lax-Wendroff type theorem for upwind finite volume schemes in $2$-D*, East-West J. Numer. Math.**4**(1996), no. 4, 279–292. MR**1430241** - N. N. Kuznecov,
*Stable methods for the solution of a first order quasilinear equation in a class of discontinuous functions*, Dokl. Akad. Nauk SSSR**225**(1975), no. 5, 1009–1012 (Russian). MR**445103** - Randall J. LeVeque,
*Numerical methods for conservation laws*, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. MR**1153252**, DOI 10.1007/978-3-0348-8629-1 - Peter Lax and Burton Wendroff,
*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**120774**, DOI 10.1002/cpa.3160130205 - Sebastian Noelle,
*Convergence of higher order finite volume schemes on irregular grids*, Adv. Comput. Math.**3**(1995), no. 3, 197–218. MR**1325031**, DOI 10.1007/BF02431999 - Stanley Osher and Richard Sanders,
*Numerical approximations to nonlinear conservation laws with locally varying time and space grids*, Math. Comp.**41**(1983), no. 164, 321–336. MR**717689**, DOI 10.1090/S0025-5718-1983-0717689-8 - Richard Sanders,
*On convergence of monotone finite difference schemes with variable spatial differencing*, Math. Comp.**40**(1983), no. 161, 91–106. MR**679435**, DOI 10.1090/S0025-5718-1983-0679435-6

## Bibliographic Information

**Volker Elling**- Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02906
- Email: velling@stanfordalumni.org
- 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.
- © Copyright 2006 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
- MathSciNet review: 2261020