  We construct a new finite difference method for the flow of ideal viscous
isentropic gas in one spatial dimension. For the continuity equation, the
method is a standard upwind discretization. For the momentum equation, the
method is an uncommon upwind discretization, where the moment and the velocity
are solved on dual grids. Our main result is convergence of the method as
discretization parameters go to zero. Convergence is proved by adapting the
mathematical existence theory of Lions and Feireisl to the numerical setting.
