Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Numerical stability for some equations of gas dynamics
HTML articles powered by AMS MathViewer

by A. Y. le Roux PDF
Math. Comp. 37 (1981), 307-320 Request permission

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.
References
  • 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 430536, DOI 10.1512/iumj.1977.26.26029
  • James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
  • 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
  • P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
  • Peter D. Lax, The formation and decay of shock waves, Visiting scholars’ lectures (Texas Tech Univ., Lubbock, Tex., 1970/71), Math. Ser., No. 9, Texas Tech Press, Texas Tech Univ., Lubbock, Tex., 1971, pp. 107–139. MR 0367471
  • A. Y. Le Roux, Approximation de Quelques Problèmes Hyperboliques Non Linéaires, Thèse d’Etat, Rennes, 1979.
  • 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 442449
  • A. Y. Le Roux, Stabilité de Schémas Numériques Adaptés à Certains Modèles Océaniques Non Linéaires, Rapport C.N.E.X.O. 78, 1963 (1980).
  • 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
  • A. Y. le Roux, Stability of numerical schemes solving quasilinear wave equations, Math. Comp. 36 (1981), no. 153, 93–105. MR 595044, DOI 10.1090/S0025-5718-1981-0595044-X
  • Tai Ping Liu, Uniqueness of weak solutions of the Cauchy problem for general $2\times 2$ conservation laws, J. Differential Equations 20 (1976), no. 2, 369–388. MR 393871, DOI 10.1016/0022-0396(76)90114-5
  • Takaaki Nishida and Joel Smoller, Mixed problems for nonlinear conservation laws, J. Differential Equations 23 (1977), no. 2, 244–269. MR 427852, DOI 10.1016/0022-0396(77)90129-2
  • 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
  • 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.
Similar Articles
Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Math. Comp. 37 (1981), 307-320
  • MSC: Primary 76N15; Secondary 65M10, 76A60
  • DOI: https://doi.org/10.1090/S0025-5718-1981-0628697-8
  • MathSciNet review: 628697