The inviscid fluxes in Eqn. Eqn_3D-NS-FV are computed using upwind approximations based on Roe's Flux-difference Splitting method [44] . In this approach the interface flux is written as the exact solution to an approximate Riemann problem, e.g.

\begin{equation} \mathbf{\hat{F}}_{i+1/2} = \frac{1}{2}\left[\left(\mathbf{\hat{F}}(\mathbf{q}^L_{i+1/2}) + \mathbf{\hat{F}}(\mathbf{q}^R_{i+1/2})\right) + \epsilon|\tilde{\mathbf{A}}|_{i+1/2}\left(\mathbf{q}^L_{i+1/2} - \mathbf{q}^R_{i+1/2}\right)\right] \end{equation}

where \(\mathbf{q}^L_{i+1/2}\) and \(\mathbf{q}^R_{i+1/2}\) are the primitive state variables to the left and right of the cell interface, and \(\tilde{\mathbf{A}} = \tilde{\mathbf{A}}(\mathbf{q}^L_{i+1/2}, \mathbf{q}^R_{i+1/2})\) denotes the Jacobian \(\partial \mathbf{\hat{F}} / \partial \mathbf{Q}\) evaluated with Roe-averaged variables. \(\epsilon\) is the entropy fix of Harten [15] .