Bv Solutions of the Semidiscrete Upwind Scheme

We consider the semidiscrete upwind scheme (0.1) u(t, x)t + 1 ( f ( u(t, x) ) − f ( u(t, x− ) )) = 0. We prove that if the initial data ū of (0.1) has small total variation, then the solution u (t) has uniformly bounded BV norm, independent of t, . Moreover by studying the equation for a perturbation of (0.1) we prove the Lipschitz continuous dependence of u (t) on the initial data. Using a technique similar to the vanishing viscosity case, we show that as → 0 the solution u (t) converges to a weak solution of the corresponding hyperbolic system, (0.2) ut + f(u)x = 0. 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.