An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws
Breuss, Michael
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005), p. 965-993 / Harvested from Numdam
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We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered and non-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 viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient are prone to oscillations at data extrema. For all LFt schemes except for the classical Lax-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-staggered version may yield oscillatory results, while it can be shown rigorously that the staggered NT scheme does not produce oscillations when using the classical minmod-limiter under a restriction on the time step size. Note that this restriction is not the same 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.

Publié le : 2005-01-01
DOI : https://doi.org/10.1051/m2an:2005042
Classification:  35L65,  65M06,  65M12 
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     author = {Breuss, Michael},
     title = {An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {39},
     year = {2005},
     pages = {965-993},
     doi = {10.1051/m2an:2005042},
     mrnumber = {2178569},
     zbl = {1077.35089},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2005__39_5_965_0}
}

Breuss, Michael. An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) pp. 965-993. doi : 10.1051/m2an:2005042. http://gdmltest.u-ga.fr/item/M2AN_2005__39_5_965_0/

[1] M. Breuß, The correct use of the Lax-Friedrichs method. ESAIM: M2AN 38 (2004) 519-540. | Numdam | Zbl 1077.65089

[2] L. Evans, Partial Differential Equations. American Mathematical Society (1998). | MR 1625845 | Zbl 0902.35002

[3] E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Edition Marketing (1991). | MR 1304494 | Zbl 0768.35059

[4] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer Verlag, New York (1996). | MR 1410987 | Zbl 0860.65075

[5] A. Harten, On a class of high order resolution total variation stable finite difference schemes. SIAM J. Numer. Anal. 21 (1984) 1-23. | Zbl 0547.65062

[6] G.-S. Jiang and E. Tadmor, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892-1917. | Zbl 0914.65095

[7] G.-S. Jiang, D. Levy, C.T. Lin, S. Osher and E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Numer. Anal. 35 (1998) 2147-2168. | Zbl 0920.65053

[8] S. Jin and Z. Xin, The relaxation scheme for systems of conservation laws in arbitrary space dimension. Comm. Pure Appl. Math. 45 (1995) 235-276. | Zbl 0826.65078

[9] P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical approximation. Comm. Pure Appl. Math. 7 (1954) 159-193. | Zbl 0055.19404

[10] P.G. Lefloch and J.-G. Liu, Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions. Math. Comp. 68 (1999) 1025-1055. | Zbl 0915.35069

[11] R.J. Leveque, Numerical Methods for Conservation Laws. Birkhäuser Verlag, 2nd Edition (1992). | MR 1153252 | Zbl 0847.65053

[12] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040

[13] D. Levy, G. Puppo and G. Russo, Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22 (2000) 656-672. | Zbl 0967.65089

[14] X.D. Liu and E. Tadmor, Third order nonoscillatory central schemes for hyperbolic conservation laws. Numer. Math. 79 (1998) 397-425. | Zbl 0906.65093

[15] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-436. | Zbl 0697.65068

[16] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp. 68 (1984) 1025-1055. | Zbl 0587.65058

[17] H. Tang and G. Warnecke, A note on (2k+1)-point conservative monotone schemes. ESAIM: M2AN 38 (2004) 345-358. | Numdam | Zbl 1075.65113