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

Convergence is proved and error bounds are derived for a finite-difference approximation to discontinuous solutions of the Navier-Stokes equations for nonisentropic, compressible flow in one space dimension. The scheme is fully implicit and can be implemented under reasonable mesh 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 initial 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 is discontinuous. This choice is dictated by the known continuous-dependence theory and accounts for the low rate of convergence in the second case.