A cell-centered diffusion scheme on two-dimensional unstructured meshes

We propose a new cell-centered diffusion scheme on unstructured meshes. The main feature of this scheme lies in the introduction of two normal fluxes and two temperatures on each edge. A local variational formulation written for each corner cell provides the discretization of the normal fluxes. This discretization yields a linear relation between the normal fluxes and the temperatures defined on the two edges impinging on a node. The continuity of the normal fluxes written for each edge around a node leads to a linear system. Its resolution allows to eliminate locally the edge temperatures as function of the mean temperature in each cell. In this way, we obtain a small symmetric positive definite matrix located at each node. Finally, by summing all the nodal contributions one obtains a linear system satisfied by the cell-centered unknowns. This system is characterized by a symmetric positive definite matrix. We show numerical results for various test cases which exhibit the good behavior of this new scheme. It preserves the linear solutions on a triangular mesh. It reduces to a classical five-point scheme on rectangular grids. For non orthogonal quadrangular grids we obtain an accuracy which is almost second order on smooth meshes.