Finite difference method for a general fractional porous medium equation

TL;DR

This paper develops and analyzes a finite difference numerical method for solving a fractional porous medium equation, proving its convergence and providing a specific approximation for the fractional derivative.

Contribution

It introduces a new finite difference scheme for fractional porous medium equations, with proven existence, uniqueness, and convergence results, including a novel two-point approximation for the fractional derivative.

Findings

Proved existence and uniqueness of the numerical solution.

Established convergence with order depending on c3.

Proposed a d-point approximation with order O(h^{2-c3}).

Abstract

We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[ \frac{\partial u}{\partial t}+(-\Delta)^{\sigma/2} (u^m)=0 \] posed for $x\in \mathbb{R}^N$, $t>0$, with $m\geq 1$, $\sigma \in (0,2)$, and nonnegative initial data $u(x,0)$. We prove existence and uniqueness of the solution of the numerical method and also the convergence to the theoretical solution of the equation with an order depending on $\sigma$. We also propose a two points approximation to a $\sigma$-derivative with order $O(h^{2-\sigma})$.