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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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