Diffusion Discretization Stencil

The flux at a given face, for example the + k -face,

$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$$_+kⁿ⁺¹ = - D_c,+k$$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$$$\Phi^{{n+1}}_{}$$ + $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{J_{+k}}$}$$

is defined using this stencil:

in the Asymmetric Method. The Support Operator Method uses all seven unknowns within a cell to define the face flux.

Each cell has a cell-centered conservation equation which involves all six face fluxes, and gives a stencil which includes all seven unknowns within the cell (in both methods).

To close the system, an equation relating the fluxes on each side of a face is added for every face in the problem. This gives the following stencil:

in the Asymmetric Method. The Support Operator Method uses all thirteen unknowns within a cell-cell pair to define the face equation.