Finite difference method for a fractional porous medium equation

Abstract

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 $x\in {\mathbb{R}}^N$, $m\ge 1$ 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.

28 pages, 9 figures

References in corpus (2)

• A PDE approach to fractional diffusion in general domains: a priori error analysis
• Classical solutions for a logarithmic fractional diffusion equation

Cited by in corpus (4)

• Finite difference method for a general fractional porous medium equation
• The Mesa Problem for the Fractional Porous Medium Equation
• The Fisher-KPP equation with nonlinear fractional diffusion
• Transformations of Self-Similar Solutions for porous medium equations of fractional type