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.

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

Article copyright:
© Copyright 1981
American Mathematical Society