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Comparison to Morel-Hall Asymmetric Method

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

FTfAf = - Df$\displaystyle {\frac{{\left( \phi_f - \phi_c \right)}}{{\left\vert \mathbf{r}_f - \mathbf{r}_c \right\vert}}}$Af  .

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:

$\displaystyle \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.$$\displaystyle \sum_{n}^{}$D-1nVnPTnJn-1J-TnPn$\displaystyle \left.\vphantom{ \sum_n D^{-1}_n V_n {\mathbf{P}}^{\mathrm{T}}_n
\mathbf{J}_n^{-1} {\mathbf{J}}^{-\mathrm{T}}_n
\mathbf{P}_n }\right]$$\displaystyle \widetilde{{\mathbf{F}}}$ = $\displaystyle \widetilde{{\mathbf{\Phi}}}$  .

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:

FTfAf = - Df$\displaystyle \left[\vphantom{ {\mathbf{J}}^{-\mathrm{T}} \mathbf{P}_f \widetilde{\mathbf{\Phi}} }\right.$J-TPf$\displaystyle \widetilde{{\mathbf{\Phi}}}$$\displaystyle \left.\vphantom{ {\mathbf{J}}^{-\mathrm{T}} \mathbf{P}_f \widetilde{\mathbf{\Phi}} }\right]^{{{\mathrm{T}}}}_{{}}$Af  .

Michael L. Hall