# The correct use of the Lax–Friedrichs method

- Volume: 38, Issue: 3, page 519-540
- ISSN: 0764-583X

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topBreuß, Michael. "The correct use of the Lax–Friedrichs method." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.3 (2004): 519-540. <http://eudml.org/doc/244814>.

@article{Breuß2004,

abstract = {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.},

author = {Breuß, Michael},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {conservation laws; numerical methods; finite difference methods; central methods; Lax–Friedrichs method; total variation stability; Lax-Friedrichs method},

language = {eng},

number = {3},

pages = {519-540},

publisher = {EDP-Sciences},

title = {The correct use of the Lax–Friedrichs method},

url = {http://eudml.org/doc/244814},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Breuß, Michael

TI - The correct use of the Lax–Friedrichs method

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 3

SP - 519

EP - 540

AB - 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.

LA - eng

KW - conservation laws; numerical methods; finite difference methods; central methods; Lax–Friedrichs method; total variation stability; Lax-Friedrichs method

UR - http://eudml.org/doc/244814

ER -

## References

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- [4] P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical approximation. Comm. Pure Appl. Math. 7 (1954) 159–193. Zbl0055.19404
- [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. Zbl0915.35069
- [6] R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser Verlag, 2nd edn. (1992). Zbl0847.65053MR1153252
- [7] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). Zbl1010.65040MR1925043
- [8] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–436. Zbl0697.65068

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