A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction. Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers. The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil. Namely, the scheme results in a global sparse diffusion matrix, which couples only the cell-centered unknowns. The space discretization 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-face normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures. The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system. This system is obtained by enforcing the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical test cases.