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

Breuss, Michael

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 5, pp. 965-993.

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.

Classification : 35L65, 65M06, 65M12
Mots clés : conservation laws, numerical methods, finite difference methods, central schemes

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

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

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

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

[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

[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

[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

[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

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

[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

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

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

[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

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

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

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

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