A Numerical Method for Porous Medium Equation by An Energetic Variational Approach

We study the numerical methods for porous media equation (PME). There are two characteristics: the finite speed of 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 the dissipative law of the total energy \(f\log f\), we obtain the trajectory equation, and then construct a fully discrete scheme. We prove that the numerical scheme is uniquely solvable on a convex set, keeps the discrete energy dissipation law and can be solved by a Damped Newton iteration method. Next, based on the dissipation law of the energy \(1/f\), we construct a linear scheme for the corresponding trajectory equation, which also keeps the discrete dissipation law. Meanwhile, under some smoothness assumption, it is proved, by a higher order expansion technique, that both schemes are all second-order convergence in space and first-order convergence in time. Each scheme yields a good approximation for the solution and the finite speed of propagation. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problems could be treated naturally, which is hard for all the existence methods. As a linear scheme, the second scheme is more efficient.

Event Topic

Stochastic & Multiscale Modeling and Computation