next up previous
Next: Algebraic Solution Up: Diffusion Previous: Diffusion Discretization Method Properties

Diffusion Discretization Stencil

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

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

is defined using this stencil:
\includegraphics[angle=-90,scale=.7]{/home/hall/Caesar/documents/images/Augustus/stencil.ps}
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).

\includegraphics[angle=-90,scale=.5]{/home/hall/Caesar/documents/images/Augustus/celleq.ps}
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:
\includegraphics[angle=-90,scale=.5]{/home/hall/Caesar/documents/images/Augustus/faceeq2.ps}
in the Asymmetric Method. The Support Operator Method uses all thirteen unknowns within a cell-cell pair to define the face equation.


next up previous
Next: Algebraic Solution Up: Diffusion Previous: Diffusion Discretization Method Properties
Michael L. Hall