A discretization method for the diffusion equation in 3-D has been developed. The method is valid for unstructured meshes with cell-centered data. The homogeneous solution of the diffusion equation, which is linear, is preserved exactly. The method is second order accurate and conserves energy locally. Material discontinuities are handled rigorously. In the case of an orthogonal mesh, the method reduces to the standard seven-point operator.

The discretization scheme results in an unsymmetric matrix with a size of roughly four times the number of cells. This matrix system can be solved using any sparse unsymmetric matrix solver.

A code package named AUGUSTUS has been developed to implement the method. This package models all of the geometries listed in Table 1. The matrix is solved using either Krylov subspace methods or an unstructured multi-frontal method.