Orthogonal Reduction

In the case of an orthogonal mesh, the face area vector for each face is parallel with the major direction for that face, and perpendicular to the minor directions for that face. The dot product of the flux vector with the area vector for a given face is therefore only related to the flux in the major direction and not the minor directions.

There is no loss of generality in assuming that the k, l, and m directions are aligned with the x, y, and z directions. With this assumption, the J^-T matrix reduces to the following: $$\mathbf{J}^{-T} = \frac{1}{J_k^x J_l^y J_m^z} \left[ \begin... ...^x J_m^z & 0 \\ 0 & 0 & J_k^x J_l^y \end{array} \right] \; .$$

The flux-area dot product for each face then becomes directly proportional to the difference in intensity values between the face and the cell center. For example, the flux dot product for the + k face in the case of an orthogonal mesh is given by¹

$$\begin{eqnarray}\html{eqn58} \lefteqn{\mbox{$\stackrel{^{\mathstrut}\smash{\lon... ... \left( \Phi^{n+1}_{+k} - \Phi^{n+1}_{c1} \right) \nonumber \; . \end{eqnarray}$$

Similarly, for the - k face of cell c2, $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c2,... ...ft( \Phi^{n+1}_{c2} - \Phi^{n+1}_{-k} \right) \nonumber \; .$$
If cell c1 and cell c2 share the + k/- k face, then these two equations may be substituted into the cell face equation, Equation 19. Noting that $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c2,-k}}$}$ = - $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c1,-k}}$}$, $$\frac{D_{c1,+k}}{J_{c1,k}^x / 2} \left( \Phi^{n+1}_{+k} - \P... ...^x / 2} \left( \Phi^{n+1}_{c2} - \Phi^{n+1}_{+k} \right) \; .$$
This equation can be solved for $\Phi^{{n+1}}_{{+k}}$ and then substituted back into Equation 23 to yield an equation for the flux which only involves the cell center unknowns:

$$\begin{eqnarray}\html{eqn75} \lefteqn{\mbox{$\stackrel{^{\mathstrut}\smash{\long... ... \left( \Phi^{n+1}_{c2} - \Phi^{n+1}_{c1} \right) \; .\nonumber \end{eqnarray}$$

This equation shows that the effective diffusion coefficient on an orthogonal mesh is $$D_{\textit{\scriptsize eff},f} = \left[ \frac{\Delta x_{c1}... ...right]^{-1} \left( \Delta x_{c1} + \Delta x_{c2} \right) \; ,$$
where J_k^x has been denoted with the more familiar form of $\Delta$x. The flux expression then becomes

$$\begin{eqnarray}\html{eqn78} \lefteqn{\mbox{$\stackrel{^{\mathstrut}\smash{\long... ... \left( \Phi^{n+1}_{c2} - \Phi^{n+1}_{c1} \right) \; , \nonumber \end{eqnarray}$$

which agrees exactly with the standard seven-point diffusion operator. It is only possible to reduce the flux expression to the cell center variables, using an effective diffusion coefficient to represent the material discontinuity, if the mesh is orthogonal. Otherwise, the gradients on each side of the interface must be represented separately, and the equation set cannot be reduced to a cell center difference using local operations.