BV solutions of the semidiscrete upwind scheme

We consider the semidiscrete upwind scheme u(t, x), + 1/ε (f(u(t, x)) - f(u(t, x - ε))) = 0. (1) We prove that if the initial data ū of (1) has small total variation, then the solution uε(t) has uniformly bounded BV norm, independent of t, ε. Moreover by studying the equation for a perturbation of (1) we prove the Lipschitz-continuous dependence of uε(t) on the initial data. Using a technique similar to the vanishing-viscosity case, we show that as ε → 0 the solution uε(t) converges to a weak solution of the corresponding hyperbolic system, ut + f(u)x, = 0. (2) Moreover this weak solution coincides with the trajectory of a Riemann semigroup, which is uniquely determined by the extension of Liu's Riemann solver to general hyperbolic systems.