BV Solutions of the Semidiscrete Upwind Scheme

We consider the semidiscrete upwind scheme

We prove that if the initial data ū of (1) has small total variation, then the solution u ^ɛ(t) has uniformly bounded BV norm, independent of t, ɛ. Moreover by studying the equation for a perturbation of (1) we prove the Lipschitz-continuous dependence of u ^ɛ(t) on the initial data. Using a technique similar to the vanishing-viscosity case, we show that as ɛ→0 the solution u ^ɛ(t) converges to a weak solution of the corresponding hyperbolic system,

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.