Nonlinear finite volume discretization for transient diffusion problems on general meshes

Abstract

A nonlinear discrete duality finite volume scheme is proposed for time-dependent diffusion equations. The model example is written in a new formulation giving rise to similar nonlinearities for both the diffusion and the potential functions. A natural finite volume discretization is built on this particular problem's structure. The fluxes are generically approximated thanks to a key fractional average. The point of this strategy is to promote coercivity and scheme's stability simultaneously. The existence of positive solutions is guaranteed. The theoretical convergence of the nonlinear scheme is established using practical compactness tools. Numerical results are performed in order to highlight the second order accuracy of the methodology and the positiveness of solutions on distorted meshes.