Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Stability of numerical schemes solving quasilinear wave equations


Author: A. Y. le Roux
Journal: Math. Comp. 36 (1981), 93-105
MSC: Primary 65M10; Secondary 35L67
DOI: https://doi.org/10.1090/S0025-5718-1981-0595044-X
MathSciNet review: 595044
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A generalization of the Riemann invariants for quasi-linear wave equations of the type $ {\partial ^2}w/\partial {t^2} = \partial f(\partial w/\partial x)/\partial x$, which includes the shock curves, is proposed and is used to solve the Riemann problem. Three numerical schemes, whose accuracy is of order one (the Lax-Friedrichs scheme and two extensions of the upstreaming scheme), are constructed by $ {L^2}$-projection, onto piecewise constant functions, of the solutions of a set of Riemann problems. They are stable in the $ {L^\infty }$-norm for a class of wave equations, including a nonlinear model of extensible strings, which are not genuinely nonlinear. The problem with boundary conditions is detailed, as is its treatment, by the numerical schemes.


References [Enhancements On Off] (What's this?)

  • [1] J. Glimm, "Solutions in the large for nonlinear hyperbolic systems of equations," Comm. Pure Appl. Math., v. 18, 1965, pp. 697-715. MR 0194770 (33:2976)
  • [2] B. L. Keyfitz & H. C. Kranzer, "Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two non linear conservation laws," J. Differential Equations, v. 27, 1978, pp. 444-476. MR 0466993 (57:6866)
  • [3] A. Y. Le Roux, Approximation de Quelques Problèmes Hyperboliques Non Linéaires, Thèse, Rennes, 1979.
  • [4] A. Y. Le Roux, "Convergence of an accurate scheme for quasilinear equations," R.A.I.R.O. (To appear.)
  • [5] A. Y. Le Roux, "Stabilité numérique de modèles océaniques non linéaires," C. R. Acad. Sci. Paris, v. 290, 1980, pp. 885-888.
  • [6] A. Y. Le Roux, Stabilité de Schéma Numériques Adaptés à Certains Modèles Océaniques Non Linéaires, Contract Report-CNEXO, 78.1963, Mars 1980.
  • [7] A. Y. Le Roux, "Numerical stability for some equations of gas dynamics." (To appear.)
  • [8] T. P. Liu, "Uniqueness of weak solutions of the Cauchy problem for general $ 2 \times 2$ conservation laws," J. Differential Equations, v. 20, 1976, pp. 369-388. MR 0393871 (52:14678)
  • [9] T. P. Liu, "The entropy condition and the admissibility of shocks," J. Math. Anal. Appl., v. 53, 1976, pp. 78-88. MR 0387830 (52:8669)
  • [10] T. P. Liu, "Existence and uniqueness theorems for Riemann problems," Trans. Amer. Math. Soc., v. 212, 1975, pp. 375-382. MR 0380135 (52:1036)
  • [11] T. Nishida, "Global solutions for an initial boundary value problem of a quasilinear hyperbolic system," Proc. Japan Acad., v. 44, 1968, pp. 642-648. MR 0236526 (38:4821)
  • [12] T. Nishida & J. A. Smoller, "Solutions in the large for some non linear hyperbolic conservation laws," Comm. Pure Appl. Math., v. 26, 1973, pp. 183-200. MR 0330789 (48:9126)
  • [13] T. Nishida & J. A. Smoller, "Mixed problem for non linear conservation laws," J. Differential Equations, v. 23, 1977, pp. 244-269. MR 0427852 (55:882)
  • [14] J. Oleinik, "Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equaton," Amer. Math. Soc. Transl., (2), v. 33, 1963, pp. 285-290.
  • [15] J. A. Smoller, "On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems," Michigan Math. J., v. 16, 1969, pp. 201-210. MR 0247283 (40:552)
  • [16] J. A. Smoller, "A uniqueness theorem for Riemann problems," Arch. Rational Mech. Anal., v. 33, 1969, pp. 110-115. MR 0237961 (38:6238)
  • [17] B. Wendroff, "The Riemann problem for materials with nonconvex equation of state. I. Isentropic flow," J. Math. Anal. Appl., v. 38, 1977, pp. 454-466; "II. General flow," J. Math. Anal. Appl., v. 38, 1977, pp. 640-658. MR 0328387 (48:6729)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M10, 35L67

Retrieve articles in all journals with MSC: 65M10, 35L67


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0595044-X
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society