Cell Face Equations

After discretizing the conservation equation, there is an equation for every cell in the problem, but roughly three extra unknowns per cell have been added. Closure is achieved by applying a continuity of flux condition at each cell face: $$- \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c... ...tackrel{^{\mathstrut}\smash{\longrightarrow}}{A_{c2,f}}$} = 0$$
where c1 and c2 are the two cells that share the face f. If, for example, face f is a - m face in c1 and is a + k face in c2, this equation can be written out as

$$\begin{eqnarray}\html{eqn47} D_{c1,-m} \; \mathbf{J}^{-T}_{c1} \left[ \begin{a... ...athstrut}\smash{\longrightarrow}}{A_{c2,+k}}$} = 0 \; . \nonumber \end{eqnarray}$$

The 11-point stencil for the cell face equations is shown in Figure 6.

Figure 6: Stencil for the face equation, shown with a dotted blue line.