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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
MathSciNet review: 595044
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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.


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  • [1] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770
  • [2] Barbara L. Keyfitz and Herbert C. Kranzer, Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws, J. Differential Equations 27 (1978), no. 3, 444–476. MR 0466993
  • [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] 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
  • [9] Tai Ping Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), no. 1, 78–88. MR 0387830
  • [10] Tai Ping Liu, Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc. 212 (1975), 375–382. MR 0380135, 10.1090/S0002-9947-1975-0380135-2
  • [11] Takaaki Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad. 44 (1968), 642–646. MR 0236526
  • [12] Takaaki Nishida and Joel A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 26 (1973), 183–200. MR 0330789
  • [13] Takaaki Nishida and Joel Smoller, Mixed problems for nonlinear conservation laws, J. Differential Equations 23 (1977), no. 2, 244–269. MR 0427852
  • [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. 16 (1969), 201–210. MR 0247283
  • [16] J. A. Smoller, A uniqueness theorem for Riemann problems, Arch. Rational Mech. Anal. 33 (1969), 110–115. MR 0237961
  • [17] Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454–466. MR 0328387

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