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# Non-reflecting boundary conditions for compressible Navier-Stokes equations

I have some questions about the implementation of non-reflecting OUTFLOW boundary condition for Navier Stokes equations.

Following

Poinsot, Lele "Boundary Conditions for Direct Simulations of Compressible Viscous Flows"

Pirozzoli, Colonius "Generalized characteristic relaxation boundary conditions for unsteady compressible flow simulations"

both authors suggest an equation like

\begin{equation}

\frac{d\mathbf{u}_b}{dt} + \mathbf{d(u_b)} + \mathcal{\mathbf{T(u_b)}} = \mathbf{S(u_b)}\quad\quad (1)

\end{equation}

where $\mathbf{u}_b = (\rho, \rho u, \rho v, \rho w, \rho e)$ are the conservative variable at the bound b; $\mathbf{d(u_b)}$ is a certain characteristic treatment of x-flux; $\mathcal{\mathbf{T(u_b)}}$ are the transverse therms; $\mathbf{S(u_b)}$ are the source therm.

Now consider a finite difference method and suppose that the bound is located at $n+1/2$ (as usual in this problems); considering furthermore $n+1, n+2,\dots, n+GN$ ghost points.

Equation (1