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Numerical instabilities in upwind methods: Analysis and cures for the “carbuncle” phenomenon. (English) Zbl 0990.76051
Summary: Some upwind formulations promote severe instabilities that originate in the numerical capturing of shocks; this is known as the “carbuncle” phenomenon. An analysis of the linearized form of the algorithms is carried out to explain and predict the generation of such instabilities. The information obtained is then used to design remedies that only slightly and locally modify the original schemes.

76M12 Finite volume methods applied to problems in fluid mechanics
76J20 Supersonic flows
76L05 Shock waves and blast waves in fluid mechanics
Full Text: DOI
[1] Charrier, P.; Dubrocca, B.; Flandrin, L., An approximate Riemann solver of hypersonic bidimensional flows, C. R. acad. sci. Paris, 317, 1083, (1993)
[2] F. Coquel, and, M. S. Liou, Field by field hybrid upwind splitting methods, AIAA Paper 93-3302-CP, 1993.
[3] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. numer. anal., 25, 294, (1988) · Zbl 0642.76088
[4] Einfeldt, B.; Munz, C.D.; Roe, P.L.; Sjögreen, B., On Godunov-type methods near low density, J. comput. phys., 92, 273, (1991) · Zbl 0709.76102
[5] Godunov, S.K., A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, 271, (1959) · Zbl 0171.46204
[6] Harten, A.; Lax, P.D.; Van Leer, B., On upstream and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35, (1983) · Zbl 0565.65051
[7] D. Hänel, R. Schwane, and G. Seider, On the accuracy of upwind schemes for the solution of the Navier-Stokes equations, AIAA Paper 87-1105, in Proc. AIAA 8th Computational Fluid Dynamics Conference, 1987, pp. 42-46.
[8] Harten, A.; Hymax, J.M., Self adjusting grid methods for one-dimensional hyperbolic conservation laws, J. comput. phys., 50, 235, (1983) · Zbl 0565.65049
[9] Kim, K.H.; Lee, J.H.; Rho, O.H., An improvement of AUSM schemes by introducing the pressure-based weight functions, Comput. fluids, 27, 311, (1998) · Zbl 0964.76064
[10] Korte, J.J., An explicit upwind algorithm for solving the parabolized Navier-Stokes equations, (February, 1991)
[11] Liou, M.S., On a new class of flux splitting, 414, (1993)
[12] Liou, M.S.; Steffen, C.J., A new flux splitting scheme, J. comput. phys., 107, 23, (1993) · Zbl 0779.76056
[13] Liou, M.S., A sequel to AUSM: AUSM+, J. comput. phys., 129, 364, (1996) · Zbl 0870.76049
[14] M. S. Liou, Probing numerical fluxes: Mass flux, positivity and entropy-satisfying property, AIAA Paper 97-2035, 1997.
[15] Osher, S.; Solomon, F., Upwind schemes for hyperbolic systems of conservation laws, Math. comput., 38, 339, (1982) · Zbl 0483.65055
[16] Pandolfi, M., A contribution to the numerical prediction of unsteady flows, Aiaa j., 22, 602, (1984) · Zbl 0542.76090
[17] Quirk, J.J., A contribution to the great Riemann solver debate, (1992) · Zbl 0794.76061
[18] Radespiel, R.; Kroll, N., Accurate flux vector splitting for shocks and shear layers, J. comput. phys., 121, 66, (1991) · Zbl 0843.76059
[19] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[20] Sanders, R.; Morano, E.; Druguet, M.-C., Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics, J. comput. phys., 145, 511, (1998) · Zbl 0924.76076
[21] Steger, J.L.; Warming, R.F., Flux splitting of the inviscid gasdynamic equations with application to finite-difference methods, J. comput. phys., 40, 263, (1981) · Zbl 0468.76066
[22] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock waves, 4, 25, (1994) · Zbl 0811.76053
[23] J. van Keuk, J. Ballmann, A. Schneider, and, W. Koschel, Numerical simulation of hypersonic inlet flows, in, Proc. of the AIAA 8th International Space Plane and Hypersonic Systems and Technologies Conference, Norfolk VA, 1998.
[24] Van Leer, B., Flux-vector splitting for the Euler equations, 170, (1982)
[25] Wada, Y.; Liou, M.S., A flux splitting scheme with high-resolution and robustness for discontinuities, (1994)
[26] Wada, Y.; Liou, M.S., An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM J. sci. comput., 18, 633, (1997) · Zbl 0879.76064
[27] Walder, R., Some aspects of the computational dynamics ofcolliding flows in astrophysical nebulae, (1993)
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