Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

   
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [BL97] A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of conservation laws, Arch. Rat. Mech. Anal. 140 (1997), 301-317. MR 1489317 (98m:35125)
  • [CCL94] B. Cockburn, F. Coquel, and P. LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), 77-103. MR 1240657 (95d:65078)
  • [CL91] F. Coquel and P. 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 (91m:65229)
  • [CL93] -, 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 (94e:65092)
  • [CM80] M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1-21.MR 0551288 (81b:65079)
  • [CT80] M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. AMS 78 (1980), no. 3, 385-390.MR 0553381 (81a:47054)
  • [DB00] J. K. Dukowicz and J. R. Baumgardner, Incremental remapping as a transport/advection algorithm, J. Comput. Phys. 160 (2000), 318-335.
  • [DK87] J. K. Dukowicz and J. W. Kodis, Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations, J. Comput. Phys. 8 (1987), no. 3, 305-321.MR 0883773 (88b:76008)
  • [Duk84] J. K. Dukowicz, Conservative rezoning (remapping) for general quadrilateral meshes, J. Comput. Phys. 54 (1984), 411-424.
  • [Ella] V. Elling, A Lax-Wendroff type theorem for semidiscrete schemes on unstructured quasi-uniform grids, in preparation.
  • [Ellb] -, Methods and theory for conservative remapping, in preparation.
  • [Ell00] -, Numerical simulation of gas flow in moving domains, Diploma Thesis, RWTH Aachen (Germany), 2000.
  • [Ell03] -, A possible counterexample to entropy solutions and to Godunov scheme convergence, 75 (2006), 1721-1733.
  • [GR96] E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Springer, 1996.MR 1410987 (98d:65109)
  • [Gra99] J. Grandy, Conservative remapping and region overlays by intersecting arbitrary polyhedra, J. Comput. Phys. 148 (1999), 433-466.MR 1669715 (99i:76114)
  • [HHL76] A. Harten, J.M. Hyman, and P.D. Lax, On finite-difference approximation and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297-321. MR 0413526 (54:1640)
  • [Jon99] P.W. Jones, First- and second-order conservative remapping schemes for grids in spherical coordinates, Monthly Weather Review (1999), no. 9, 2204-2210.
  • [Kru70] S.N. Kruzkov, First order quasi-linear equations in several independent variables, Mat. Sb. 81 (1970), no. 2, 285-355, transl. in Math. USSR Sb. 10 (1970) no. 2, 217-243. MR 0267257 (42:2159)
  • [KRW96] 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), 279-292. MR 1430241 (97m:65158)
  • [Kuz75] N.N. Kuznetsov, On stable methods for solving a first-order quasi-linear equation in the class of discontinuous functions, Dokl. Akad. Nauk. SSSR 225 (1975), no. 5, 25-28, transl. in USSR Comp. Math. and Math. Phys. 16 (1976) no. 6, 105-119. MR 0445103 (56:3448)
  • [LeV92] R.J. LeVeque, Numerical methods for conservation laws, 2nd ed., Birkhäuser, 1992. MR 1153252 (92m:65106)
  • [LW60] P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217-237. MR 0120774 (22:11523)
  • [Noe95] S. Noelle, Convergence of higher order finite volume schemes on irregular grids, Adv. Comp. Math. 3 (1995), 197-218.MR 1325031 (96a:65137)
  • [OS83] S. Osher and R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comp. 41 (1983), no. 164, 321-336. MR 0717689 (85i:65121)
  • [San83] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91-106.MR 0679435 (84a:65075)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M12, 35L65

Retrieve articles in all journals with MSC (2000): 65M12, 35L65


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

American Mathematical Society