Weak solutions and convergent numerical schemes of modified compressible Navier–Stokes equations

Lately, there has been some interest in modifications of the compressible Navier–Stokes equations to include diffusion of mass. In this paper, we investigate possible ways to add mass diffusion to the 1-D Navier–Stokes equations without violating the basic entropy inequality. As a result, we recover Brenner's modification of the Navier–Stokes equations as a special case. We consider Brenner's system along with another modification where the viscous terms collapse to a Laplacian diffusion. For each of the two modifications, we derive a priori estimates for the PDE, sufficiently strong to admit a weak solution; we propose a numerical scheme and demonstrate that it satisfies the same a priori estimates. For both modifications, we then demonstrate that the numerical schemes generate solutions that converge to a weak solution (up to a subsequence) as the grid is refined.