An explicit finite-difference scheme for one-dimensional Generalized Porous Medium Equations: Interface tracking and the hole filling problem

Monsaingeon, Léonard

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1011-1033

Based on results of E. DiBenedetto and D. Hoff we propose an explicit finite-difference scheme for one dimensional Generalized Porous Medium Equations ${\partial }_{t}u={\partial }_{xx}^2\varphi \left(u\right)$. The scheme allows to track the moving free boundaries, and captures the so-called hole filling phenomenon when free boundaries collide. We prove the convergence of the discrete solution when the mesh parameter $\Delta x\to 0$. Finally, we provide numerical evidence of the convergence of the scheme.

Classification : 65M06, 35R35, 35K65
Keywords: Generalized porous medium equation, interface tracking, hole filling, finite-difference

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