A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
Provides a convergent numerical scheme for a class of PDEs relevant to porous media flow, but the contribution is incremental as it extends existing hybrid finite volume methods to degenerate parabolic problems.
The authors propose a finite volume method for degenerate parabolic convection-reaction-diffusion equations on general meshes, achieving convergence via a priori estimates and compactness. The scheme handles anisotropic diffusion and is robust for high Péclet numbers.
We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection-reaction-diffusion equation. Equations of this type arise in many contexts, such as the modeling of contaminant transport in porous media. We discretize the diffusion term, which can be anisotropic and heterogeneous, via a hybrid finite volume scheme. We construct a partially upwind scheme for the convection term. We consider a wide range of unstructured possibly non-matching polygonal meshes in arbitrary space dimension. The only assumption on the mesh is that the volume elements must be star-shaped. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet--Kolmogorov compactness theorem.