A Local Support-Operators Diffusion Discretization Scheme for Hexahedral Meshes

We derive a cell-centered 3-D diffusion differencing scheme for unstructured hexahedral meshes using the local support-operators method. Our method is said to be local because it yields a sparse matrix representation for the diffusion equation, whereas the traditional support-operators method yields a dense matrix representation. The diffusion discretization scheme that we have developed offers several advantages relative to existing schemes. Most importantly, it offers second-order accuracy on reasonably well-behaved nonsmooth meshes, rigorously treats material discontinuities, and has a symmetric positive-definite coefficient matrix. The order of accuracy is demonstrated computationally rather than theoretically. Rigorous treatment of material discontinuities implies that the normal component of the flux is continuous across such discontinuities while the parallel components may be either continuous or discontinuous in accordance with the exact solution to the problem being considered. The only disadvantage of the method is that it has both cell-centered and face-centered scalar unknowns as opposed to just cell-center scalar unknowns. Computational examples are given which demonstrate the accuracy and cost of the new scheme.