Conservation Equation

The first step in the derivation of the method is the discretization of the conservation equation, Equation 2. Integrating the conservation equation over the cell volume gives

$$\begin{eqnarray}\html{eqn11} \int_{V_c} \alpha \frac{\partial \Phi}{\partial t}... ... + \int_{V_c} \sigma \Phi dV = \int_{V_c} S dV \; . \nonumber \end{eqnarray}$$

Defining cell averages and applying Gauss' Theorem gives $$\alpha_c \frac{\partial \Phi_c}{\partial t} V_c + \int_{A} \m... ...\longrightarrow}}{dA}$} + \sigma_c \Phi_c V_c = S_c V_c \; .$$
Discretizing temporally and evaluating the flux integral gives

$$\begin{eqnarray}\html{eqn14} \frac{\alpha_c V_c}{\Delta t} \left( \Phi_c^{n+1} ... ...A_f}$} \\ {}+\sigma_c \Phi_c^{n+1} V_c = S_c V_c \; , \nonumber \end{eqnarray}$$

where the sum over f represents the sum over the face values, the subscript c represents a cell-centered or cell-average value and the unsuperscripted variables are evaluated at n + ${\frac{{1}}{{2}}}$. Note that all of the geometries in Table 1 can be represented by this equation when the proper definitions for areas, volumes, and face sums are substituted. Note that this scheme is not limited to fully implicit differencing; for instance, a Crank-Nicholson differencing scheme may be employed.