Discretization Strategy

Cell-Center Equations - Integrate the Diffusion Equation over a cell, and temporally discretize via Backwards Euler:

$$\alpha^{n}_{}$$$${\frac{{\left( \Phi^{n+1} - \Phi^n \right)}}{{\Delta t}}}$$V_c + $$\sum_{f}^{}$$$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$$ ^. $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c,f}}$}$$

+  $$\sigma_{c}^{n}$$$$\Phi_{c}^{{n+1}}$$V_c = S_cⁿV_c

Face Equations - Flux continuity at every face:

$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c1,f1}^{n+1}}$}$$ ^. $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c1,f1}}$}$$ = - $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c2,f2}^{n+1}}$}$$ ^. $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c2,f2}}$}$$

Boundary Face Equations - Robin condition:

$$\beta^{1}_{c}$$$$\Phi_{{c,f}}^{}$$ - $$\beta^{2}_{c}$$$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$$ ^. $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c,f}}$}$$ = $$\beta^{3}_{c}$$$$\Phi_{{b}}^{}$$

Note that:

• This discretization will be inherently conservative.
• No derivatives are taken across material boundaries - a rigorous treatment.
• $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$ ^. $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c,f}}$}$ remains to be defined (in terms of $\Phi$ ).

Unknowns for $\Phi$ are located at the cell centers and the cell faces. Cell Equations will involve these seven unknowns:

Face Equations will involve the thirteen unknowns from two adjacent cells:

This gives a local stencil in terms of cell-center and cell-face unknowns.

The $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$ ^. $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c,f}}$}$ terms, on each face of a cell, must still be defined in terms of the $\Phi$ 's within that cell.