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An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws

Published online by Cambridge University Press:  15 September 2005

Michael Breuss
Michael Breuss*
Affiliation:
Technische Universität Braunschweig, Computational Mathematics, Pockelsstraße 14, 38106 Braunschweig, Germany.
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Abstract

We discuss the occurrence of oscillationswhen using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered andnon-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema.The dependence of oscillatory properties on the numerical viscositycoefficient is investigated rigorously for the LFt schemes, illuminating alsothe properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient areprone to oscillations at data extrema. For all LFt schemes except for the classicalLax-Friedrichs method, occurring oscillations are damped in the course of a computation.This damping effect also holds for Rusanov's method. Concerning the NT schemes, the non-staggeredversion may yield oscillatory results, while it can be shown rigorously that the staggered NTscheme does not produce oscillations when using the classical minmod-limiter under arestriction on the time step size. Note that this restriction is not thesame as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar problems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper.

Keywords

Conservation lawsnumerical methodsfinite difference methodscentral schemes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 965 - 993
Copyright
© EDP Sciences, SMAI, 2005

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References

Breuß, M., The correct use of the Lax-Friedrichs method. ESAIM: M2AN 38 (2004) 519–540. CrossRef
L. Evans, Partial Differential Equations. American Mathematical Society (1998).
E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Edition Marketing (1991).
E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer Verlag, New York (1996).
Harten, A., On a class of high order resolution total variation stable finite difference schemes. SIAM J. Numer. Anal. 21 (1984) 1–23. CrossRef
Jiang, G.-S. and Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892–1917. CrossRef
Jiang, G.-S., Levy, D., Lin, C.T., Osher, S. and Tadmor, E., High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Numer. Anal. 35 (1998) 2147–2168. CrossRef
Jin, S. and Xin, Z., The relaxation scheme for systems of conservation laws in arbitrary space dimension. Comm. Pure Appl. Math. 45 (1995) 235–276. CrossRef
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 Edition (1992).
R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).
Levy, D., Puppo, G. and Russo, G., Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22 (2000) 656–672. CrossRef
Liu, X.D. and Tadmor, E., Third order nonoscillatory central schemes for hyperbolic conservation laws. Numer. Math. 79 (1998) 397–425. CrossRef
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–436. CrossRef
Tadmor, E., Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp. 68 (1984) 1025–1055.
Tang, H. and Warnecke, G., A note on (2k + 1)-point conservative monotone schemes. ESAIM: M2AN 38 (2004) 345–358. CrossRef