The correct use of the Lax–Friedrichs method

Michael Breuß

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Breuß, 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

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  1. L. Evans, Partial Differential Equations. American Mathematical Society (1998).  Zbl0902.35002
  2. E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Ellipses, Edition Marketing (1991).  
  3. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).  Zbl0860.65075
  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).  Zbl0847.65053
  7. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).  Zbl1010.65040
  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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