A CONVERGENT FINITE-DIFFERENCE SCHEME FOR THE NAVIER-STOKES EQUATIONSOF ONE-DIMENSIONAL, NONISENTROPIC, COMPRESSIBLE FLOW

Convergence is proved and error bounds are derived for a finite-differ ence approximation to discontinuous solutions of the Navier-Stokes equ ations for nonisentropic, compressible flow in one space dimension. Th e scheme is fully implicit and can be implemented under reasonable mes h conditions. It is shown that the approximations converge at the rate O(Delta x(1/2)) when the initial data is in H-1, and O(Delta x(a)) (a < 1\12) when the initial velocity and energy are in L(2), and the ini tial density is piecewise H-1. The errors are measured in a norm that dominates the sup-norm of the error in the density, which in general i s discontinuous. This choice is dictated by the known continuous-depen dence theory and accounts for the low rate of convergence in the secon d case.