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

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Abstract: A generalization of the Riemann invariants for quasi-linear wave equations of the type , 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 -projection, onto piecewise constant functions, of the solutions of a set of Riemann problems. They are stable in the -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.

**[1]**James Glimm,*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**0194770**, https://doi.org/10.1002/cpa.3160180408**[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**, https://doi.org/10.1016/0022-0396(78)90062-1**[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**, https://doi.org/10.1016/0022-0396(76)90114-5**[9]**Tai Ping Liu,*The entropy condition and the admissibility of shocks*, J. Math. Anal. Appl.**53**(1976), no. 1, 78–88. MR**0387830**, https://doi.org/10.1016/0022-247X(76)90146-3**[10]**Tai Ping Liu,*Existence and uniqueness theorems for Riemann problems*, Trans. Amer. Math. Soc.**212**(1975), 375–382. MR**0380135**, https://doi.org/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**, https://doi.org/10.1002/cpa.3160260205**[13]**Takaaki Nishida and Joel Smoller,*Mixed problems for nonlinear conservation laws*, J. Differential Equations**23**(1977), no. 2, 244–269. MR**0427852**, https://doi.org/10.1016/0022-0396(77)90129-2**[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**, https://doi.org/10.1007/BF00247755**[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**, https://doi.org/10.1016/0022-247X(72)90103-5

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0595044-X

Article copyright:
© Copyright 1981
American Mathematical Society