A convergent finite-difference method for a nonlinear variational wave equation

We establish rigorously convergence of a semidiscrete upwind scheme for the nonlinear variational wave equation u(tt) - c(u)(c(u)u(x))(x) = 0 with u vertical bar(t= 0) = u(0) and u(t)vertical bar(t=0) = upsilon(0). Introducing Riemann invariants R = u(t) + cu(x) and S = u(t) - cu(x), the variational wave equation is equivalent to R-t - cR(x) = (c) over tilde (R-2 - S-2) and S-t + cS(x) = -(c) over tilde (R-2 - S-2) with (c) over tilde = c'/(4c). An upwind scheme is defined for this system. We assume that the speed c is positive, increasing and both c and its derivative are bounded away from zero and that R vertical bar(t=0), S vertical bar(t=0) is an element of L-1(R) boolean AND L-3(R) are nonpositive. The numerical scheme is illustrated on several examples.