On the Discretization of Diffusion Fluxes for a System of PDEs

We consider a system of two PDEs with diffusion type fluxes and discontinuous coefficients. Extending on the ideas of the simple diffusion problem, we derive a two-point flux approximation for a cell centered finite volume method with minimal stencil on a regular Cartesian grid. The proposed approximation takes the coupled characteristic of the original problem into account and is compared to a commonly deployed decoupled approximation using separate harmonic averages of the transport coefficients. Equivalence of both methods is shown if the coefficients fulfill some linear relation and generalization to non-uniform rectilinear grids is discussed. We conclude our analysis by performing a numerical study for our specific application of liquid electrolytes in Li-ion batteries.