Stability of numerical schemes solving quasilinear wave equations

le Roux, A. Y.

doi:10.1090/S0025-5718-1981-0595044-X

Stability of numerical schemes solving quasilinear wave equations
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by A. Y. le Roux PDF

Math. Comp. 36 (1981), 93-105 Request permission

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

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
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 466993, DOI 10.1016/0022-0396(78)90062-1

Approximation de Quelques Problèmes Hyperboliques Non Linéaires

R.A.I.R.O.

C. R. Acad. Sci. Paris

Stabilité de Schéma Numériques Adaptés à Certains Modèles Océaniques Non Linéaires

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
Tai Ping Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), no. 1, 78–88. MR 387830, DOI 10.1016/0022-247X(76)90146-3
Tai Ping Liu, Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc. 212 (1975), 375–382. MR 380135, DOI 10.1090/S0002-9947-1975-0380135-2
Takaaki Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad. 44 (1968), 642–646. MR 236526
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 330789, DOI 10.1002/cpa.3160260205
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

Amer. Math. Soc. Transl.

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 247283, DOI 10.1307/mmj/1029000262
J. A. Smoller, A uniqueness theorem for Riemann problems, Arch. Rational Mech. Anal. 33 (1969), 110–115. MR 237961, DOI 10.1007/BF00247755
Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454–466. MR 328387, DOI 10.1016/0022-247X(72)90103-5