SYMMETRIC POSITIVE DEFINITE FLUX-CONTINUOUS FULL-TENSOR FINITE-VOLUME SCHEMES ON UNSTRUCTURED CELL-CENTERED TRIANGULAR GRIDS

摘要

Novel cell-centered full-tensor finite-volume methods are presented for general unstructured grids in two spatial dimensions. The numerical schemes are flux-continuous and based on computing the transmissibilities in a local subcell transform space, ensuring that local flux matrices are symmetric. As a result the global discretization matrix is shown to be symmetric positive definite for any grid type. A symmetric physical space method is also introduced, and the symmetric methods are shown to be closely related. Discrete ellipticity conditions are derived for positive definiteness of the physical space and subcell space schemes. Computational examples are presented for unstructured triangular grids demonstrating good performance of the scheme. The schemes are compared with the so-called multipoint flux approximation (MPFA) O-method [I. Aavatsmark, T. Barkve, O. Boe, and T. Mannseth, SIAM J. Sci. Comput., 19 (1998), pp. 1700–1716]. Good agreement between the methods is obtained, but the new scheme shows improved behavior in challenging cases.