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Abstract: We establish rigorously convergence of a semi-discrete upwind scheme for thenonlinear variational wave equation $u {tt} - cucu u x x = 0$ with$u| {t=0}=u 0$ and $u t| {t=0}=v 0$. Introducing Riemann invariants $R=u t+cu x$ and $S=u t-c u x$, the variational wave equation is equivalent to $R t-cR x=\tilde c R^2-S^2$ and $S t+c S x=-\tilde c R^2-S^2$ with $\tildec=c-4c$. An upwind scheme is defined for this system. We assume that the thespeed $c$ is positive, increasing and both $c$ and its derivative are boundedaway from zero and that $R| {t=0}, S| {t=0}\in L^1\cap L^3$ are nonpositive.The numerical scheme is illustrated on several examples.



Autor: H. Holden, K. H. Karlsen, N. H. Risebro

Fuente: https://arxiv.org/







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