Finite volume inflow/outflow BCs

Niels

Hello,

I am trying to figure out what boundary conditions to apply at inflow and outflow boundaries for my 3D finite volume solver. I came across this old paper:

Ph. Rostand, B. Stoufflet, "Finite Volume Galerkin Methods for Viscous Gas Dynamics", INRIA, 1988 http://hal.inria.fr/inria-00075691/en/

Section II.3 mentions a mechanism whereby such BC's can be imposed by means of a split of the inviscid flux Jacobian A = dF/dU according to the sign of the eigenvalues:

A = A+ + A-

where

A+ = R(D+)L, A- = R(D-)L

and D+, D- contain only the non-negative and non-positive eigenvalues respectively. I realize that for a homogenous function of degree 1, as is the case for the Euler flux function, the fluxes can now be calculated as

F(U) = (A+)U + (A-)U

However, the paper says that the boundary fluxes at an inflow or outflow can be calculated as:

F(U) = (A-)U_external + (A+)U_internal

U_external is the state exterior to the computational domain, U_internal the state just inside the boundary and where the flux Jacobians are evaluated at U_internal. I don't understand. Intuitively, I would have it this way:

F(U) = (A-)U_internal + (A+)U_external

because A+ and A- correspond to characteristics entering and leaving the computational domain. For example, at an inflow I would like extererior information to propagate into the domain.

I tried to find the background paper by Steger and Warming, but failed. So if anyone can explain why I am wrong, that would be great.

Thanks, Martin