A finite difference scheme for a system of two conservation laws with artificial viscosity

Abstract:

In this paper we analyze an implicit finite difference scheme for the mixed initial-value Dirichlet problem for a system of two conservation laws with artificial viscosity. The system we consider is a model for isentropic flow in one space dimension. First, we show that, under certain conditions on the mesh, the scheme is stable in the sense that it possesses an invariant set (defined by the so-called Riemann invariants). We obtain this result as an extension of the same stability theorem for the Lax-Friedrichs scheme in the inviscid case. Second, we show that the approximants remain bounded and, in fact, decay to the boundary values as $t \to \infty$. Finally, we obtain two $O(\Delta {x^2})$ error bounds; the first grows exponentially in time while the second, which requires that the data have small oscillation, is independent of time.