# Finite volume scheme for a parabolic equation with a non-monotone diffusion function

Evans and Portilheiro introduced in 2004 the functional framework that allows to tackle

the problem of a forward-backward diffusion equation with a cubic-like diffusion function,

that is classically ill-posed. The key is to consider its ``entropy'' formulation

determined by considering the equation as the singular limit of a third-order

pseudo-parabolic equation. Obtaining numerical simulations is not easy, since

the ill-posedness related to the negativity of the diffusion coefficient induces

severe oscillations. However, we showed that, in 1D, the regularization offered by

the basic Euler in time-centered finite differences in space renders a fairly

good numerical solution, except for the fact that the entropy condition is

violated. We thus proposed an adapted entropic scheme in 1D. The finite volume framework

has since allowed us to prove new properties of the problem.

- Last modified: 16/05/2016