Finite difference method for a fractional porous medium equation
This provides a numerical scheme for a class of nonlinear fractional diffusion equations, but the results are incremental as they extend existing methods to a specific equation.
The authors develop a finite difference method for the fractional porous medium equation using the Caffarelli-Silvestre extension, proving existence, uniqueness, and convergence, with numerical experiments on typical initial data.
We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[\frac{\partial u}{\partial t}+(-Δ)^{1/2} (u^m)=0.\] 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.