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A general, non-iterative Riemann solver for Godunov’s method. (English) Zbl 0629.76074

Godunov’s method is characterized by the use of a Riemann problem solution to resolve discontinuities at the interface between cells. The major drawback of this method is the difficulty and the high cost of solving the (nonlinear) Riemann problem exactly, especially for materials with complex equations of state. This paper describes a simplified and noniterative approximate Riemann solver which is characterized by only two material-dependent parameters. For a given material, these parameters are the local speed of sound and a parameter which is directly related to the shock density ratio in the limit of strong shocks. These parameters are conveniently obtained from a linear fit to the experimental data for the shock Hugoniot in various materials. The approximate Riemann solver retains the essential quadratic nonlinearity which enables it to deal with the whole range of cases from weak sound waves to strong shocks.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76M99 Basic methods in fluid mechanics
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[1] von Neumann, J.; Richtmyer, R.D., J. appl. phys., 21, 232, (1950)
[2] Godunov, S.K., Mat. sh., 47, 271, (1959)
[3] Richtmyer, R.D.; Morton, K.W., Difference methods for initial value problems, (1967), Interscience New York · Zbl 0155.47502
[4] Roe, P.L., ()
[5] Godunov, S.K.; Zabrodin, A.V.; Prokopov, G.P., USSR comput. math. math. phys., 1, 1187, (1961), (Engl. Transl.)
[6] Enquist, B.; Osher, S., Math. comput., 34, 45, (1980)
[7] Roe, P.L., J. comput. phys., 43, 357, (1981)
[8] Pandolfi, M., Aiaa j., 22, 602, (1984)
[9] van Leer, B., J. comput. phys., 32, 101, (1979)
[10] Collela, P.; Woodward, P.R., J. comput. phys., 54, 174, (1984)
[11] Woodward, P.R.; Colella, P., J. comput. phys., 54, 115, (1984)
[12] Wilkins, M.L., J. comput. phys., 36, 281, (1980)
[13] Dukowicz, J.K., J. comput. phys., 54, 411, (1984)
[14] Thompson, P.A., Compressible-fluid dynamics, (1972), McGraw-Hill New York · Zbl 0251.76001
[15] Landshoff, R., A numerical method for treating fluid flow in the presence of shocks, Los alamos scientific laboratory report LA-1930, (1955), Los Alamos, N.M.
[16] Courant, R.; Friedrichs, K.O., Supersonic flow and shock waves, (1948), Interscience New York · Zbl 0041.11302
[17] Colella, P.; Glaz, H.M., Efficient solution algorithms for the Riemann problem for real gases, Lawrence Berkeley laboratory report LBL-15776, (1983), Berkeley, Calif.
[18] Colella, P., SIAM J. sci. statist, comput., 3, 76, (1982)
[19] Marsh, S.P., LASL shock hugoniot data, (1980), Univ. of California Press Berkeley/Los Angeles
[20] Sod, G.A., J. comput. phys., 27, 1, (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.