Finite difference method for a fractional porous medium equation

TL;DR

This paper develops and analyzes a numerical finite difference method for solving a fractional porous medium equation involving the fractional Laplacian, demonstrating convergence and validating with numerical experiments.

Contribution

It introduces a novel finite difference scheme for the fractional porous medium equation and proves its convergence and uniqueness of solutions.

Findings

The method converges to the theoretical solution.

Existence and uniqueness of the numerical solution are established.

Numerical experiments confirm the method's effectiveness.

Abstract

We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[\frac{\partial u}{\partial t}+(-\Delta)^{1/2} (u^m)=0.\] The problem is posed in $x\in \mathbb{R}^N$, $m\geq 1$ and with nonnegative initial data. The fractional Laplacian is implemented via the so-called Caffarelli-Silvestre extension. We prove existence and uniqueness of the solution of this method and also the convergence to the theoretical solution of the equation. We run numerical experiments on typical initial data.