Numerical methods for Porous Medium Equation by an Energetic Variational Approach

TL;DR

This paper develops and analyzes two energetic variational numerical schemes for the porous media equation, effectively handling free boundary propagation, waiting time phenomena, and ensuring convergence and stability.

Contribution

It introduces two novel energetic variational schemes for PME, one nonlinear and one linear, with proven convergence, dissipation preservation, and effective treatment of free boundary and waiting time issues.

Findings

Both schemes are second-order in space and first-order in time.

Schemes preserve discrete dissipation laws.

The linear scheme is more computationally efficient.

Abstract

We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on $f \log f$ as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on $\frac{1}{2 f}$ as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, it is proved, by a higher order expansion technique, that both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existence methods. Due to its linear nature, the second scheme is more efficient.