## 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, - 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, - 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,"

*Approximation de Quelques Problèmes Hyperboliques Non Linéaires*, Thèse d’Etat, Rennes, 1979.

*Stabilité de Schémas Numériques Adaptés à Certains Modèles Océaniques Non Linéaires*, Rapport C.N.E.X.O. 78, 1963 (1980).

*J. Math. Pures Appl.*, v. 38, 1977, pp. 454-466; II. General now,

*J. Math. Pures Appl.*, v. 38, 1977, pp. 640-658.

## 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