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Computations of compressible multifluids. (English) Zbl 1033.76029
Summary: Recent years have seen a growing interest in developing numerical algorithms for compressible multifluids. Computations ran into unexpected difficulties due to oscillations generated at material interfaces, and understanding of the underlying mechanisms was needed before these oscillations could be circumvented. This paper reviews some of the recent models and numerical algorithms that have been proposed, and points to key ideas that they have in common. Noting the known fact that such oscillations do not arise in single-fluid computations, an extremely simple algorithm is proposed which circumvents the oscillations and amounts to computing two different flux functions across material fronts, to update the different fluids on both sides.

76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
76T30 Three or more component flows
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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[1] Abgrall, R., Generalisation of the ROE scheme for the computation of mixture of perfect gases, Rech. Aérosp, 6, (1988)
[2] Abgrall, R., How to prevent pressure oscillations in multicomponent flows: A quasi conservative approach, J. comput. phys., 125, 150, (1996) · Zbl 0847.76060
[3] R. Abgrall, B. N’Konga, and, R. Saurel, Efficient numerical approximation of compressible multi-material flow for unstructured meshes, submitted for publication. · Zbl 1084.76543
[4] Chargy, D.; Abgrall, R.; Fezoui, L.; Larrouturou, B., Conservative numerical schemes for multicomponent inviscid flows, Rech. Aérosp., 2, 61, (1992)
[5] Chern, I.-L.; Glimm, J.; McBryan, O.; Plohr, B.; Yaniv, S., Front tracking for gas dynamics, J. comput. phys., 62, 83, (1986) · Zbl 0577.76068
[6] Clarke, J.F.; Karni, S.; Quirk, J.J.; Simmonds, L.G.; Roe, P.L.; Toro, E.F., Numerical computation of two-dimensional unsteady detonation waves in high energy solids, J. comput. phys., 106, 215, (1993) · Zbl 0770.76040
[7] Cocchi, J.-P.; Saurel, R., A Riemann problem based method for the resolution of compressible material flows, J. comput. phys., 137, 265, (1997) · Zbl 0934.76055
[8] Colella, P.; Glaz, H.M.; Ferguson, R.E., Multifluid algorithms for Eulerian finite difference methods, Unpublished, (1989)
[9] Coquel, F.; El-Almine, K.; Godlewski, E.; Perthame, B.; Rascle, P., A numerical method using upwind schemes for the resolution of two-phase flows, J. comput. phys., 136, 272, (1997) · Zbl 0893.76052
[10] Courant, R.; Friedrichs, K.O., Supersonic flow and shock waves, pure and applied mathematics, (1948) · Zbl 0041.11302
[11] Davis, S.F., An interface tracking method for hyperbolic systems of conservation laws, Appl. numer. math., 10, 447, (1992) · Zbl 0766.65067
[12] Fedkiw, R.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 457, (1999) · Zbl 0957.76052
[13] Fedkiw, R.; Aslam, T.; Xu, S., The ghost fluid method for deflagration and detonation discontinuities, J. comput. phys., 154, 393, (1999) · Zbl 0955.76071
[14] Fedkiw, R.; Marquina, A.; Merriman, B., An isobaric fix for the overheating problem in multimaterial compressible flows, J. comput. phys., 148, 545, (1999) · Zbl 0933.76075
[15] Godunov, S.K., A difference scheme for numerical computation of discontinous solutions of equations of fluids dynamics, Math. sb., 47, 271, (1959) · Zbl 0171.46204
[16] Ivings, M.J.; Causon, D.M.; Toro, E.F., On hybrid high resolution upwind methods for multicomponent flows, Z. angew. math. mech., 77, 645, (1997) · Zbl 0889.76047
[17] Ivings, M.J.; Causon, D.M.; Toro, E.F., On Riemann solvers for compressible liquids, Int. J. numer. meth. fluids, 3, 395, (1998) · Zbl 0918.76047
[18] Jenny, P.; Mueller, B.; Thomann, H., Correction of conservative Euler solvers for gas mixtures, J. comput. phys., 132, 91, (1997) · Zbl 0879.76059
[19] Karni, S., Multi-component flow calculations by a consistent primitive algorithm, J. comput. phys., 112, 31, (1994) · Zbl 0811.76044
[20] Karni, S., Hybrid multifluid algorithms, SIAM J. sci. comput., 17, 1019, (1996) · Zbl 0860.76056
[21] S. Karni, Compressible bubbles with surface tension, in 16th International Conference on Numerical Methods in Fluid Dynamics, edited by Ch.-H. BruneauxSpringer-Verlag, Berlin/New York, 1998, pp. 506-512.
[22] S. Karni, A level-set scheme for compressible interfaces, in Numerical Methods for Wave Propagation, edited by E. F. Toro and J. F. ClarkeKluwer Academic Publishers, Dordrecht, 1988, pp. 253-274.
[23] Larouturou, B., How to preserve the mass fraction positive when computing compressible multi-component flows, J. comput. phys., 95, 59, (1991)
[24] Larouturou, B.; Fezoui, L., On the equations of multicomponent perfect or real gas inviscid flow, Lecture notes in mathematics, 1402, 69-97, (1989)
[25] Miller, G.H.; Puckett, E.G., A high order Godunov method for multiple condensed phases, J. comput. phys., 128, 134, (1996) · Zbl 0861.65117
[26] Mulder, W.; Osher, S.; Sethian, J., Computing interface motion: the compressible rayleigh – taylor and kelvin – helmholtz instabilities, J. comput. phys., 100, 209, (1992) · Zbl 0758.76044
[27] Noh, W.F.; Woodward, P.R., Slic (simple line interface calculation), Springer lecture notes in physics, 59, 330-339, (1976) · Zbl 0382.76084
[28] Quirk, J.J.; Karni, S., On the dynamics of a shock – bubble interaction, J. fluid mech., 318, 129, (1996) · Zbl 0877.76046
[29] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[30] P. L. Roe, Fluctuations and signals—A framework for numerical evolution problems, in Numerical Methods for Fluid Dynamics, edited by K. W. Morton and M. J. BainesAcademic Press, New York, 1982, pp. 219-257.
[31] Roe, P.L., A new approach to computing discontinuous flows of several ideal gases, (1984)
[32] Saurel, R.; Abgrall, R., A simple method for compressible multifluid flows, SIAM J. sci. comp., 21, (1999) · Zbl 0957.76057
[33] Saurel, R.; Abgrall, R., Some models and methods for compressible multifluid and multiphase flows, J. comput. phys., 150, 425, (1999) · Zbl 0937.76053
[34] Shyue, K.-M., An efficient shock-capturing algorithm for compressible multicomponent problems, J. comput. phys., 1, 208, (1998) · Zbl 0934.76062
[35] Shyue, K.-M., A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state, J. comput. phys., 1, 43, (1999) · Zbl 0957.76039
[36] Ton, V., Improved shock-capturing methods for multicomponet and reacting flows, J. comput. phys., 128, 237, (1996) · Zbl 0860.76060
[37] E. F. Toro, Primitive, conservative and adaptive schemes for hyperbolic conservation laws, in Numerical Methods for Wave Propagation (Manchester 95), edited by E. F. Toro and J. F. ClarkeKluwer Academic Publishers, Dordrecht, 1998, pp. 253-274.
[38] van Leer, B., Towards the ultimate conservative difference scheme: 2. monotonicity and conservation combined in a second-order scheme, J. comput. phys., 14, 361, (1974) · Zbl 0276.65055
[39] Xu, K., BGK-based scheme for multicomponent flow calculations, J. comput. phys., 1, 122, (1998) · Zbl 0882.76060
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