Abstract: 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|_t=0=u₀ and u_t|_t=0=v₀. 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(R²−S²) and S_t+cS_x=−~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.