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Numerical stability for some equations of gas dynamics

Author: A. Y. le Roux
Journal: Math. Comp. 37 (1981), 307-320
MSC: Primary 76N15; Secondary 65M10, 76A60
MathSciNet review: 628697
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Abstract: The isentropic gas dynamics equations in Eulerian coordinates are expressed by means of the density $ \rho $ and the momentum $ q = \rho u$, instead of the velocity u, in order to get domains bounded and invariant in the $ (\rho ,q)$-plane, for a wide class of pressure laws $ p(\rho )$ and in the monodimensional case. A numerical scheme of the transport-projection type is proposed, which builds an approximate solution valued in such a domain. Since the characteristic speeds are bounded in this set, the stability condition is easily fulfilled and then estimates in the $ {L^\infty }$-norm are derived at any time step. Similar results are extended to the model involving friction and topographical terms, and for a simplified model of supersonic flows. The nonapplication of this study to the gas dynamics in Lagrangian coordinates is shown.

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  • [1] K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), no. 2, 373–392. MR 0430536
  • [2] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770
  • [3] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.) 47 (89) (1959), 271–306 (Russian). MR 0119433
  • [4] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653
  • [5] Peter D. Lax, The formation and decay of shock waves, Visiting scholars’ lectures (Texas Tech Univ., Lubbock, Tex., 1970/71), Texas Tech Press, Texas Tech Univ., Lubbock, Tex., 1971, pp. 107–139. Math. Ser., No. 9. MR 0367471
  • [6] A. Y. Le Roux, Approximation de Quelques Problèmes Hyperboliques Non Linéaires, Thèse d'Etat, Rennes, 1979.
  • [7] Alain Yves le Roux, Étude du problème mixte pour une équation quasi-linéaire du premier ordre, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 5, A351–A354 (French, with English summary). MR 0442449
  • [8] A. Y. Le Roux, Stabilité de Schémas Numériques Adaptés à Certains Modèles Océaniques Non Linéaires, Rapport C.N.E.X.O. 78, 1963 (1980).
  • [9] Alain Yves le Roux, Stabilité numérique de modèles océaniques non linéaires, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 19, A885–A888 (French, with English summary). MR 580164
  • [10] A. Y. le Roux, Stability of numerical schemes solving quasilinear wave equations, Math. Comp. 36 (1981), no. 153, 93–105. MR 595044, 10.1090/S0025-5718-1981-0595044-X
  • [11] Tai Ping Liu, Uniqueness of weak solutions of the Cauchy problem for general 2×2 conservation laws, J. Differential Equations 20 (1976), no. 2, 369–388. MR 0393871
  • [12] Takaaki Nishida and Joel Smoller, Mixed problems for nonlinear conservation laws, J. Differential Equations 23 (1977), no. 2, 244–269. MR 0427852
  • [13] Stanley Osher, Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 179–204. MR 605507
  • [14] B. Wendroff, "The Riemann problem for materials with non convex equation of state I. Isentropic flow," J. Math. Pures Appl., v. 38, 1977, pp. 454-466; II. General now, J. Math. Pures Appl., v. 38, 1977, pp. 640-658.

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Article copyright: © Copyright 1981 American Mathematical Society