Comparison to Morel-Hall Asymmetric Method

For an orthogonal grid, the flux out of a face can be defined simply as

F^T_fA_f = - D_f$${\frac{{\left( \phi_f - \phi_c \right)}}{{\left\vert \mathbf{r}_f - \mathbf{r}_c \right\vert}}}$$A_f  .

But for a skewed grid, this is incorrect.

The Support Operator Method corrects the left hand side of the equation, defining each $\phi$ difference in terms of all the face fluxes:

$$\left[\vphantom{ \sum_n D^{-1}_n V_n {\mathbf{P}}^{\mathrm{T}}_n \mathbf{J}_n^{-1} {\mathbf{J}}^{-\mathrm{T}}_n \mathbf{P}_n }\right.$$$$\sum_{n}^{}$$D^-1_nV_nP^T_nJ_n^-1J^-T_nP_n$$\left.\vphantom{ \sum_n D^{-1}_n V_n {\mathbf{P}}^{\mathrm{T}}_n \mathbf{J}_n^{-1} {\mathbf{J}}^{-\mathrm{T}}_n \mathbf{P}_n }\right]$$$$\widetilde{{\mathbf{F}}}$$ = $$\widetilde{{\mathbf{\Phi}}}$$  .

The Morel-Hall Asymmetric Method corrects the right hand side of the equation, defining each face flux in terms of all of the $\phi$ differences:

F^T_fA_f = - D_f$$\left[\vphantom{ {\mathbf{J}}^{-\mathrm{T}} \mathbf{P}_f \widetilde{\mathbf{\Phi}} }\right.$$J^-TP_f$$\widetilde{{\mathbf{\Phi}}}$$$$\left.\vphantom{ {\mathbf{J}}^{-\mathrm{T}} \mathbf{P}_f \widetilde{\mathbf{\Phi}} }\right]^{{{\mathrm{T}}}}_{{}}$$A_f  .