Well posedness of a time-difference scheme for a degenerate fast diffusion problem

We study a time-difference scheme for a nonlinear degenerate parabolic equation with a transport term. The model generally describes diffusion in porous media with the formation of a free boundary, this being expressed by the presence of a multivalued function in the equation. We consider singular boundary conditions which contain the multivalued function as well, and prove the stability and the convergence of the scheme, emphasizing the precise nature of the convergence. This approach is aimed to be a mathematical background which justifies the correctness of the numerical algorithm for computing the solution to this type of equations by avoiding the approximation of the multivalued function. The theory is illustrated by numerical results which put into evidence both the effects due to the equation degeneration and the formation and advance of the free boundary.