CONSTRUCTION AND CONVERGENCE STUDY OF SCHEMES PRESERVING THE ELLIPTIC LOCAL MAXIMUM PRINCIPLE

International audienceWe present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models

Piecewise linear transformation in diffusive flux discretization

To ensure the discrete maximum principle or solution positivity in finite volume schemes, diffusive flux is sometimes discretized as a conical combination of finite differences. Such a combination may be impossible to construct along material discontinuities using only cell concentration values. This is often resolved by introducing auxiliary node, edge, or face concentration values that are explicitly interpolated from the surrounding cell concentrations. We propose to discretize the diffusive flux after applying a local piecewise linear coordinate transformation that effectively removes the discontinuities. The resulting scheme does not need any auxiliary concentrations and is therefore remarkably simpler, while being second-order accurate under the assumption that the structure of the domain is locally layered.Comment: 11 pages, 1 figures, preprint submitted to Journal of Computational Physic

Monotone corrections for generic cell-centered Finite Volume approximations of anisotropic diffusion equations

We present a nonlinear technique to correct a general Finite Volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many Finite Volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding solutions converge to the continuous one as the size of the mesh tends to 0. Finally we present numerical results showing these corrections suppress local minima produced by the initial Finite Volume scheme

Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach