Weighted positive nonlinear finite volume method for dominated anisotropic diffusive equations

Abstract

In this paper, we propose a new positive finite volume scheme for degenerate parabolic equations with strongly anisotropic diffusion tensors. The key idea is to approximate the fluxes thanks to a weighted-centered scheme depending on the sign of the stiffness coefficients. More specifically, we employ a centered discretization for the mobility-like function when the transmissibilities are positive and a weighted harmonic scheme in regions where the negative transmissibilities occur. This technique prevents the formation of undershoots which entails the positivity-preserving of the approach. Also, the scheme construction retains the main elements, namely the coercivity and compactness estimates, allowing the existence and the convergence of the nonlinear finite volume scheme under general assumptions on the physical inputs, the nonlinearities, and the mesh. The numerical implementation of the methodology shows that the lower bound on the discrete solution is respected and optimal convergence rates are recovered on strongly anisotropic various test-cases.