An algorithm for one-dimensional Generalized Porous Medium Equations: interface tracking and the hole filling problem

Based on results of E. DiBenedetto and D. Hoff we propose an explicit finite difference scheme for the one dimensional Generalized Porous Medium Equation $\partial_t u=\partial_{xx}^2 \Phi(u)$. The scheme allows to track the moving free boundaries and captures the hole filling phenomenon when two free boundaries collide. We give an abstract convergence result when the mesh parameter $\Delta x\to 0$ without any error estimates, and invesigate numerically the convergence rates.