Boundary Conditions

Because the conservation equation only involves variables within a cell, the boundary conditions only affect the cell face equations. On the boundaries, instead of a cell face equation, a Robin boundary condition is specified, $$\beta_1 \; \Phi_f^{n+1} - \beta_2 \; \mbox{$\stackrel{^{\ma... ...mash{\longrightarrow}}{A_{c,f}}$} = \beta_3 \; \Phi_{bc} \; ,$$
where $\beta_{1}^{}$,$\beta_{2}^{}$ and $\beta_{3}^{}$ can be specified to match the following common boundary conditions:

• homogeneous: $\Phi_{f}^{{n+1}}$ = 0,
• reflective: - $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$ ^. $\hat{{n}}_{{c,f}}^{}$ = 0,
• vacuum: ${\frac{{1}}{{2}}}$$\Phi_{f}^{{n+1}}$ - $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$ ^. $\hat{{n}}_{{c,f}}^{}$ = 0,
• Dirichlet: $\Phi_{f}^{{n+1}}$ = $\Phi_{{bc}}^{}$,
• Neumann: - $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$ ^. $\hat{{n}}_{{c,f}}^{}$ = - $\Phi_{{bc}}^{}$  , or
• source boundary conditions:
  ${\frac{{1}}{{2}}}$$\Phi_{f}^{{n+1}}$ - $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F_{c,f}^{n+1}}$}$ ^. $\hat{{n}}_{{c,f}}^{}$ = ${\frac{{1}}{{2}}}$$\Phi_{{bc}}^{}$.