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Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics. I: One-dimensional flow. (English) Zbl 0842.76053
The author gives a detailed analysis of the limiting behavior of solutions of the Euler equations for compressible subsonic flow, with the Mach number approaching zero. A semi-implicit low Mach number extension of a Godunov-type MUSCL scheme for compressible flow is constructed with the aid of a single time but multiple space scales asymptotic analysis, which reveals three distinct roles of the pressure, namely its role as a thermodynamic variable, an acoustic wave amplitude, and the balance for the inertial forces in small scale flow structures. In addition to the development of the scheme, the paper contains a critical survey of the literature pertaining to the subject. It can be expected that the results of the analysis will definitely help to clarify alternative approaches in the development of solutions for incompressible flows and improve the rate of convergence for low Mach number solutions.
Reviewer: E.Krause (Aachen)

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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HLLE
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[1] Lee, J.H.S.; Moen, I.O., Prog. energy combust. sci., 6, 359, (1980)
[2] Patankar, S.V., Numerical heat transfer and fluid flow, (1980), Hemisphere New York · Zbl 0595.76001
[3] Casulli, V.; Greenspan, D., Int. J. numer. methods. fluids, 4, 1001, (1984)
[4] Patnaik, G.; Guirguis, R.H.; Boris, J.P.; Oran, E.S., J. comput. phys., 71, (1987)
[5] Merkle, C.L.; Choi, Y.-H., Aiaa j., 25, 831, (1987)
[6] Abarbanel, S.; Duth, P.; Gottlieb, D., Comput. & fluids, 17, 1, (1989)
[7] Fernandez, G., ()
[8] Zienkiewicz, O.C.; Szmeltzer, J.; Peraire, J., Comput. methods appl. mech. eng., 78, 105, (1990)
[9] Gustafsson, B.; Stoor, H., SIAM J. numer. anal., 28, 1523, (1991)
[10] Sesterhenn, J.; Müller, B.; Thomann, H., Computation of compressible low Mach number flow, (), 829
[11] van Leer, B., J. comput. phys., 14, 361, (1979)
[12] LeVeque, R.J., ()
[13] Klainerman, S.; Majda, A.J., Commun. pure appl. math., 35, 629, (1982)
[14] Majda, A.J.; Sethian, J., Combust. sci. technol., 42, 185, (1985)
[15] Hunter, J.K.; Majda, A.J.; Rosales, R.R., Stud. appl. math., 75, 187, (1986)
[16] Klein, R.; Peters, N., J. fluid mech., 187, 197, (1988)
[17] S. Schochet, preprint, School of Mathematics, Tel Aviv University, 1993.
[18] Majda, A.J.; Lamb, K., ()
[19] W.E., Commun. part. differential eqs., 17, 347, (1992)
[20] Matalon, M.; Matkowski, B.J., J. fluid mech., 124, 239, (1982)
[21] Chorin, A.J., Math. comput., 22, 745, (1968)
[22] Klein, R.; Munz, C.D., Proceedings, intl. conf. on numer. methods in cont. mech., (), to appear
[23] Schneider, W., Mathematische methoden in der strömungsmechanik, (1978), Vieweg University of Minnesota, Minneapolis · Zbl 0369.76001
[24] LeVeque, R.J., SIAM J. numer. anal., 22, 1051, (1985)
[25] Harten, A.; Lax, P.; van Leer, B., SIAM rev., 25, 35, (1983)
[26] Einfeldt, B., SIAM J. numer. anal., 25, 294, (1988)
[27] J. F. Gerbeau, N. Glinsky-Olivier, and B. Larrouturou, in preparation, 1993.
[28] Roe, P.L., J. comput. phys., 43, 357, (1981)
[29] R. Klein, and C. D. Munz, “The extension of incompressible flow solvers to the weakly compressible regime.” to be submitted. · Zbl 1042.76045
[30] Colella, P., High-resolution numerical methods for low Mach number flows, (), to appear
[31] Lange, K., Diplomarbeit (physik), (1993), Institut für Technische Mechanik RWTH Wiesbaden, (unpublished)
[32] Ebin, D.G., Commun. pure appl. math., 15, 451, (1982)
[33] Schochet, S., J. diff. equations, 75, 1, (1988)
[34] Embid, P., Commun. partial diff. equations, 12, 1227, (1987) · Zbl 0632.76075
[35] Embid, P., Commun. partial diff. equations, 19, 1249, (1989)
[36] Kreiss, H.-O., Commun. pure appl. math., 33, 399, (1980)
[37] Tadmor, E., Commun. pure appl. math., 35, 839, (1982)
[38] Majda, A.J.; Rosales, R.R.; Schönbeck, M., Stud. appl. math., 79, 205, (1988)
[39] Majda, A.J.; Rosales, R.R., SIAM J. appl. math., 47, 1017, (1987)
[40] Almgren, R.F., SIAM J. appl. math., 51, 351, (1991)
[41] van Kan, J., SIAM J. sci. stat. comput., 7, 870, (1986)
[42] Noll, B., ()
[43] Bell, J.B.; Colella, P.; Glaz, H.M., J. comput. phys., 85, 257, (1989)
[44] A. S. Almgren, J. B. Bell, P. Colella, and T. Marthaler, Lawrence Livermore Nat. Lab. Preprint UCRL-JC-118091, 1994.
[45] Munz, C.D., On the comparison and construction of two-step schemes for the Euler equations, () · Zbl 0796.76057
[46] Colella, P., SIAM J. sci. stat. comput., 6, 104, (1985)
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