Convergence of a positive nonlinear DDFV scheme for degenerate parabolic equations

Abstract

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way by an upstream scheme. The novelty of our approach is twofold: on the one hand we prove that the resulting scheme preserves the positivity and on the other hand we establish energy estimates. Some numerical tests are presented and they show that the scheme in question turns out to be robust and efficient. The accuracy is almost of second order on general meshes when the solution is smooth.

Positive Scharfetter-Gummel finite volume method for convection-diffusion equations on polygonal meshes

“…It is only first order accurate in space which is due to the additional artificial viscosity spanned by upwinding. A similar idea has been generalized to DDFV framework in [28]. It was observed that the accuracy is reduced when the solution is only continuous.…”

Nonlinear finite volume discretization for transient diffusion problems on general meshes

“…In a general context we proposed in [46] an alternative correction to the finite volume scheme so that one can recover the discrete maximum principle. Although the approach provided in [46] seems to be quite general, the scheme turns out to be accurate for smooth analytical solutions. It further exhibits low accuracy when the exact solution is merely continuous because of the upwinding.…”

“…The latter is obviously fulfilled under restrictive assumptions placed on the mesh or on the tensor Λ. In a general context we proposed in [46] an alternative correction to the finite volume scheme so that one can recover the discrete maximum principle. Although the approach provided in [46] seems to be quite general, the scheme turns out to be accurate for smooth analytical solutions.…”

Weighted positive nonlinear finite volume method for dominated anisotropic diffusive equations

“…The first objective of this work is to devise and investigate an improved positive finite volume discretization for diffusion equations by extending the ideas of [8,19]. In the present paper, the employed technique is generic on finite volume schemes that could be written using the two points like structure [11,16,28]. Here, it is applied in the context of the control volume finite element approximation [1,17].…”