M-Matrix Flux Splitting for General Full Tensor Discretization Operators on Structured and Unstructured Grids

Diagonal tensor flux approximations are commonly used in fluid dynamics. This approximation introduces an O(1) error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding. In general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the Jacobian matrix. After decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms, time-split semi-implicit, stable, full tensor flux approximations are introduced with in a general finite volume formalism, enabling the standard diagonal tensor Jacobian matrix structure to be retained for single phase flow, IMPES, and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids. Stability of the scheme is proven for constant elliptic coefficients. The results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids.