The equation to be solved is given by

which can be written

where

Everything in this equation is assumed to be known, with the exception of and at the new time step. This equation has an extra term from the standard diffusion equation, the term, which allows it to model the P

The new method shares these properties with the method described in

- 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).
- 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.

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) |