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.