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An asymptotic expansion for the solution of the generalized Riemann problem. II: Application to the equation of gas dynamics. (English) Zbl 0703.35106
[For part I, cf. the second and the third author, ibid. 5, No.2, 179-207 (1988; Zbl 0679.35064).]
This paper applies the general theory of approximation of generalized Riemann solutions due to the second and the third author to the equations of gas dynamics.
Explicit formulae are derived for the first order approximation of such solutions. This approximation allows to construct easily a second order accurate version of the numerical Godunov method.
Reviewer: A.Bourgeade

MSC:
35L65 Hyperbolic conservation laws
35C20 Asymptotic expansions of solutions to PDEs
76N15 Gas dynamics (general theory)
35A35 Theoretical approximation in context of PDEs
35L67 Shocks and singularities for hyperbolic equations
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References:
[1] Ben Artzi, M.; Falcovitz, J., A second order Godunov-type scheme for compressible fluid dynamics, J. Comp. Phys., Vol. 55, 1-32, (1984) · Zbl 0535.76070
[2] Ben Artzi, M.; Falcovitz, J., An upwind second-order scheme for compressible duct flows, Siam J. Sci. Comp., Vol. 7, 744-768, (1986) · Zbl 0594.76057
[3] Bourgeade, A.; Le Floch, Ph.; Raviart, P. A., Approximate solutions of the generalized Riemann problem and applications, Proceedings of Saint-Étienne (France), No. 1270, (1986), Springer Verlag · Zbl 0703.35106
[4] Courant, R.; Friedrichs, K. O., Supersonic flows and shock waves, Pure Appl. Math., Vol. 1, (1948), Interscience Pub. New York · Zbl 0041.11302
[5] Harabetian, E., A Cauchy kovalevska theorem for strictly hyperbolic systems of conservation laws with piecewise analytic initial data, Ph. D. dissertation, (1984), University of California Los Angeles
[6] Le Floch, Ph.; Raviart, P. A., An asymptotic expansion for the solution of the generalized Riemann problem, Part 1: General Theory, Ann. Inst. H. Poincaré, Nonlinear Analysis, Note aux C.R. Acad. Sci. Paris, T. 304, Série I, No. 4, 119-222, (1987)
[7] Le Floch, Ph., Sur l’étude théorique et l’approximation numérique des systèmes hyperboliques non linéaires, Thèse, (janvier 1988), École Polytechnique
[8] Li Tatsien and Yu Wenci, Boundary Value Problem for Quasilinear Hyperbolic Systems, Duke Univ. Math. Series, 1985.
[9] Smoller, J., Shock waves and reaction diffusion equations, (1983), Springer Verlag New York · Zbl 0508.35002
[10] Van Leer, B., Toward the ultimate conservative difference scheme, V, J. Comp. Phys., Vol. 32, 101-136, (1979) · Zbl 1364.65223
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