Existence, uniqueness, and computation of solutions for mixed problems in compressible fluid flow

Abstract

We consider initial-boundary value problems for the 1-D Navier-Stokes equations of compressible flow on a finite interval. For each of three different cases of initial and boundary data, we prove convergence of a finite difference approximation to a unique solution. For discontinuous initial data which is BV and piecewise smooth, the density remains discontinuous and the error in the approximate solutions is bounded by O(h 1 4 − ϱ ) for any ϱ > 0 in a norm which dominates the sup-norm of the density. For H 1 initial data, the error is bounded by O(h 1 2 ) in the same norm.