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

Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates