Method Overview

The equation to be solved is given by $$\alpha \frac{\partial \Phi}{\partial t} - \mbox{$\mbox{$\stac... ...box{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{J}$} \; ,$$
which can be written

$$\begin{eqnarray}\html{eqn2} \alpha \frac{\partial \Phi}{\partial t} + \mbox{$\m... ...\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{J}$} \; , \end{eqnarray}$$

where

$$\begin{eqnarray}\html{eqn4} \Phi & = & \mbox{\hspace{2em} Intensity} \\ \mbox... ...ongrightarrow}}{J}$} & = & \mbox{\hspace{2em} Flux Source Term} . \end{eqnarray}$$

Everything in this equation is assumed to be known, with the exception of $\Phi$ and $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ at the new time step. This equation has an extra term from the standard diffusion equation, the $\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{J}$}$ term, which allows it to model the P₁ and SP_N equations.

The new method shares these properties with the method described in ##more92 (##more92):

• It is cell-centered (balance equations are done over a cell).
• The method has cell-centered and face-centered unknowns, which are required to rigorously treat material discontinuities (see Figure 2).

  Figure 2: Location of the unknowns on the hexahedron: one at the cell center and one on each cell face.

  
• Homogeneous solutions (in this case, linear solutions) are preserved exactly.
• It is second-order accurate.
• The method reduces to the standard cell-centered operator (seven-point for 3-D, five-point for 2-D, three-point for 1-D) for an orthogonal mesh.
• Local energy conservation is maintained.
• Unfortunately, the method results in an unsymmetric matrix system.

In addition, the new method handles unstructured meshes and is multi-dimensional. The geometries that can be handled by the new method are listed in Table 1.

Table 1: The geometries that can be handled by the new method, all of which have an unstructured (arbitrarily connected) format.

Dimension	Geometries	Type of Elements
1-D	spherical, cylindrical or cartesian	line segments
2-D	cylindrical or cartesian	quadrilaterals or triangles
3-D	cartesian	hexahedra or degenerate hexahedra (tetrahedra, prisms, pyramids)