Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T01:35:08.510Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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