Weakly monotone finite volume scheme for parabolic equations in strongly anisotropic media

In this work, we propose an original idea consisting of preserving the weak monotonicity of the CVFE scheme, or generally for schemes written under the two-point like formulation. The setting handles highly anisotropic and heterogeneous diffusion tensors. The key idea is to insert a nonlinear correcting coefficient whose objective is to eliminate the anti-diffusive fluxes. This modification works on the same stencil as the initial discretization. The obtained scheme remains stable in the sense that the solution respects its physical ranges and enables the energy estimates. The existence of discrete solutions is also valid. The numerical section highlights the accuracy, the robustness and the efficiency of the novel scheme compared to the standard CVFE methodology. An application of the developed weakly monotone finite volume scheme to the simulation of mass transfer in hygroscopic media is conducted, with a specific focus on mass diffusion within wood.