A convergent finite difference method for a nonlinear variational wave equation

We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x{)}_x=0$ with $u{\mathrm{\mid }}_{t=0}=u_0$ and $u_t{\mathrm{\mid }}_{t=0}=v_0$. Introducing Riemann invariants $R=u_t+c{u}_x$ and $S=u_t-c{u}_x$, the variational wave equation is equivalent to $R_t-c{R}_x=\tilde{c}(R^2-S^2)$ and $S_t+c{S}_x=-\tilde{c}(R^2-S^2)$ with $\tilde{c}=c^{\mathrm{\prime }}\mathrm{/}(4c)$. An upwind scheme is defined for this system. We assume that the the speed $c$ is positive, increasing and both $c$ and its derivative are bounded away from zero and that $R{\mathrm{\mid }}_{t=0},S{\mathrm{\mid }}_{t=0}\in L^1\cap L^3$ are nonpositive. The numerical scheme is illustrated on several examples.