AUSM Mach and pressure splits

AUSM-family schemes split the face Mach number and pressure with low-order polynomials in the subsonic range and upwind steps outside the sonic point. These splits are pure and independent of mesh state.

Mach splits

With sense \(s=\pm 1\) (left/right):

\[M^{\pm}_{(1)} = \tfrac{1}{2}(M \pm |M|)\]

\[M^{\pm}_{(2)} = \pm \tfrac{1}{4}(M \pm 1)^2\]

\[\begin{split}M^{\pm}_{(4)} = \begin{cases} M^{\pm}_{(1)} & |M| \ge 1 \\ M^{\pm}_{(2)}\big(1 \mp 16\beta M^{\mp}_{(2)}\big) & |M| < 1 \end{cases}\end{split}\]

Pressure split

\[\begin{split}P^{\pm}_{(5)} = \begin{cases} \tfrac{1}{2}\big(1 \pm \mathrm{sign}(M)\big) & |M| \ge 1 \\ M^{\pm}_{(2)}\big((2\mp M) \mp 16\alpha M M^{\mp}_{(2)}\big) & |M| < 1 \end{cases}\end{split}\]

Pure APIs: FMSplit1, FMSplit2, FMSplit4, FPSplit5 (NsInvFluxPolicy.h). Tests: inv_flux_policy_test.