The correct use of the Lax-Friedrichs method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, p. 519-540

We are concerned with the structure of the operator corresponding to the Lax-Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax-Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version of the initial data cannot increase by the application of the scheme. The usefulness of the recipe is confirmed by numerical tests.

DOI : https://doi.org/10.1051/m2an:2004027
Classification:  35L65,  65M06,  65M12
Keywords: conservation laws, numerical methods, finite difference methods, central methods, Lax-Friedrichs method, total variation stability
@article{M2AN_2004__38_3_519_0,
author = {Breu\ss , Michael},
title = {The correct use of the Lax-Friedrichs method},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {3},
year = {2004},
pages = {519-540},
doi = {10.1051/m2an:2004027},
zbl = {1077.65089},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_3_519_0}
}

Breuß, Michael. The correct use of the Lax-Friedrichs method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, pp. 519-540. doi : 10.1051/m2an:2004027. http://www.numdam.org/item/M2AN_2004__38_3_519_0/

[1] L. Evans, Partial Differential Equations. American Mathematical Society (1998). | MR 1625845 | Zbl 0902.35002

[2] E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Ellipses, Edition Marketing (1991). | MR 1304494 | Zbl 0768.35059

[3] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). | MR 1410987 | Zbl 0860.65075

[4] P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical approximation. Comm. Pure Appl. Math. 7 (1954) 159-193. | Zbl 0055.19404

[5] P.G. Lefloch and J.-G. Liu, Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions. Math. Comp. 68 (1999) 1025-1055. | Zbl 0915.35069

[6] R.J. Leveque, Numerical Methods for Conservation Laws. Birkhäuser Verlag, 2nd edn. (1992). | MR 1153252 | Zbl 0847.65053

[7] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040

[8] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-436. | Zbl 0697.65068