A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids (Communications in Computational Physics, 2014)
We present a finite volume based cell-centered method for solving diffusionequations on three-dimensional unstructured grids with general tensor conduction.Our main motivation concerns the numerical simulation of the coupling between fluidflows and heat transfers. The corresponding numerical scheme is characterized bycell-centered unknowns and a local stencil. Namely, the scheme results in a globalsparse diffusion matrix, which couples only the cell-centered unknowns. The spacediscretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is characterized by the introduction of sub-facenormal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct anexplicit approximation of the sub-face normal heat fluxes in terms of the cell-centeredtemperature and the adjacent sub-face temperatures. The elimination of the sub-facetemperatures with respect to the cell-centered temperatures is achieved locally at eachnode by solving a small and sparse linear system. This system is obtained by enforcingthe continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration. The parallel implementation of the numericalalgorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volume method are assessed by means of various numericaltest cases.