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The correct use of the Lax–Friedrichs method

Published online by Cambridge University Press:  15 June 2004

Michael Breuß
Michael Breuß*
Affiliation:
Technical University Brunswick, Department for Analysis, Pockelsstraße 14, 38106 Brunswick, Germany. michael.breuss@math.u-bordeaux.fr.
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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 thediscussion, 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.

Keywords

Conservation lawsnumerical methodsfinite difference methodscentral methodsLax–Friedrichs methodtotal variation stability.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 3 , May 2004 , pp. 519 - 540
Copyright
© EDP Sciences, SMAI, 2004

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References

L. Evans, Partial Differential Equations. American Mathematical Society (1998).
E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Ellipses, Edition Marketing (1991).
E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical approximation. Comm. Pure Appl. Math. 7 (1954) 159–193. CrossRef
LeFloch, P.G. and Liu, J.-G., Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions. Math. Comp. 68 (1999) 1025–1055. CrossRef
R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser Verlag, 2nd edn. (1992).
R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–436. CrossRef