# The correct use of the Lax–Friedrichs method

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 38.3 (2010): 519-540. <http://eudml.org/doc/194226>.

@article{Breuß2010,

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},

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

language = {eng},

month = {3},

number = {3},

pages = {519-540},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/194226},

volume = {38},

year = {2010},

}

TY - JOUR

AU - Breuß, Michael

TI - The correct use of the Lax–Friedrichs method

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

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.; conservation laws; Lax-Friedrichs method; total variation stability

UR - http://eudml.org/doc/194226

ER -

## References

top- L. Evans, Partial Differential Equations. American Mathematical Society (1998).
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- E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
- P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical approximation. Comm. Pure Appl. Math.7 (1954) 159–193.
- P.G. LeFloch and J.-G. Liu, Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions. Math. Comp.68 (1999) 1025–1055.
- 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).
- H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys.87 (1990) 408–436.

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