Upwinding of the source term at interfaces for Euler equations with high friction

We consider Euler equations with a friction term that describe an isentropic gas flow in a porous domain. More precisely, we consider the transition between low and high friction regions. In the high friction region the system is reduced to a parabolic equation, the porous media equation. In this paper we present a hyperbolic approach based on a finite volume technique to compute numerical solutions for the system in both regimes. The Upwind Source at Interfaces (USI) scheme that we propose satisfies the following properties. Firstly it preserves the nonnegativity of gas density. Secondly, and this is the motivation, the scheme is asymptotically consistent with the limit model (porous media equation) when the friction coefficient goes to infinity. We show analytically and through numerical results that the above properties are satisfied. We shall also compare results given with the use of USI, hyperbolic–parabolic coupling and classical centered sources schemes.