A convergent finite difference method for a nonlinear variational wave equation
H. Holden, K. H. Karlsen and N. H. Risebro
Abstract: We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation utt−c(u)(c(u)ux)x=0 with u|t=0=u0 and ut|t=0=v0. Introducing Riemann invariants R=ut+cux and S=ut−cux, the variational wave equation is equivalent to Rt−cRx=~c(R2−S2) and St+cSx=−~c(R2−S2) with ~c=c′/(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|t=0, S|t=0∈L1(R)∩L3(R) are nonpositive. The numerical scheme is illustrated on several examples.