The correct use of the Lax-Friedrichs method
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 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 : 10.1051/m2an:2004027
Classification : 35L65, 65M06, 65M12
Mots-clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {519--540},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/m2an:2004027},
     zbl = {1077.65089},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2004027/}
}
TY  - JOUR
AU  - Breuß, Michael
TI  - The correct use of the Lax-Friedrichs method
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2004
SP  - 519
EP  - 540
VL  - 38
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2004027/
DO  - 10.1051/m2an:2004027
LA  - en
ID  - M2AN_2004__38_3_519_0
ER  - 
%0 Journal Article
%A Breuß, Michael
%T The correct use of the Lax-Friedrichs method
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2004
%P 519-540
%V 38
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2004027/
%R 10.1051/m2an:2004027
%G en
%F M2AN_2004__38_3_519_0
Breuß, Michael. The correct use of the Lax-Friedrichs method. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 519-540. doi : 10.1051/m2an:2004027. http://www.numdam.org/articles/10.1051/m2an:2004027/

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

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

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

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

[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

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

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

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

Cité par Sources :