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Central schemes and contact discontinuities

Published online by Cambridge University Press:  15 April 2002

Alexander Kurganov  and
Guergana Petrova
Alexander Kurganov
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. ([email protected])
Guergana Petrova
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. ([email protected])
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Abstract

We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes,proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition, suggested by Nessyahu and Tadmor [J. Comput. Phys.87 (1990) 408-463], which is efficiently applied in the context of the new schemes.The method is tested on the one-dimensional Euler equations, subject to differentinitial data, and the results are compared to the numericalsolutions, computed by other second-order central schemes.The numerical experiments clearly illustrate the advantages of theproposed technique.

Keywords

Euler equations of gas dynamicspartial characteristic decompositionfully-discrete and semi-discrete central schemes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 6 , November 2000 , pp. 1259 - 1275
Copyright
© EDP Sciences, SMAI, 2000

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References

Arminjon, P. and Viallon, M.-C., Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d'espace. C.R. Acad. Sci. Paris Sér. I 320 (1995) 85-88.
Arminjon, P., Viallon, M.-C. and Madrane, A., A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservation laws on unstructured grids. Int. J. Comput. Fluid Dyn. 9 (1997) 1-22. CrossRef
Bianco, F., Puppo, G. and Russo, G., High order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput. 21 (1999) 294-322. CrossRef
Einfeldt, B., Godunov-type, On methods for gas dynamics. SIAM J. Numer. Anal. 25 (1988) 294-318. CrossRef
Friedrichs, K.O., Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7 (1954) 345-392. CrossRef
Harten, A., The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes. Math. Comp. 32 (1978) 363-389.
Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357-393. CrossRef
Harten, A., Engquist, B., Osher, S. and Chakravarthy, S.R., Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 71 (1987) 231-303. 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
A. Kurganov, Conservation laws: stability of numerical approximations and nonlinear regularization. Ph.D. thesis, Tel-Aviv University, Israel (1997).
A. Kurganov and D. Levy, A third-order semi-discrete central scheme for conservation laws and convection-diffusion equations. SIAM J. Sci. Comput. (to appear).
A. Kurganov and G. Petrova, A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems. Numer. Math. (to appear).
A. Kurganov, S. Noelle and G. Petrova, Semi-Discrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations. SIAM J. Sci. Comput. (submitted).
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. CrossRef
Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math. 7 (1954) 159-193. CrossRef
van Leer, B., Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136. CrossRef
Levy, D., Puppo, G. and Russo, G., Central WENO schemes for hyperbolic systems of conservation laws. ESAIM: M2AN 33 (1999) 547-571. CrossRef
Levy, D., Puppo, G. and Russo, G., A third order central WENO scheme for 2D conservation laws. Appl. Numer. Math. 33 (2000) 407-414. CrossRef
Levy, D., Puppo, G. and Russo, G., Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22 (2000) 656-672. CrossRef
K.-A. Lie and S. Noelle, Remarks on high-resolution non-oscillatory central schemes for multi-dimensional systems of conservation laws. Part I: An improved quadrature rule for the flux-computation. SIAM J. Sci. Comput. (submitted).
Liu, X.-D. and Tadmor, E., Third order nonoscillatory central scheme 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) 408-463. CrossRef
Osher, S. and Tadmor, E., On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. CrossRef
Sanders, R. and Weiser, A., A high order staggered grid method for hyperbolic systems of conservation laws in one space dimension. Comput. Methods Appl. Mech. Engrg. 75 (1989) 91-107. CrossRef
Sanders, R. and Weiser, A., High resolution staggered mesh approach for nonlinear hyperbolic systems of conservation laws. J. Comput. Phys. 101 (1992) 314-329. CrossRef
Woodward, P. and Colella, P., The numerical solution of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54 (1988) 115-173. CrossRef