Finite difference method for a fractional porous medium equation
We formulate a numerical method to solve the porous medium type equation with fractional diffusion $$\begin{aligned} \frac{\partial u}{\partial t}+(-\Delta )^{1/2} (u^m)=0. \end{aligned}$$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 CaffarelliSilvestre 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 as well as a section that summarizes and concludes the proposed method.
finite difference numerical method fractional laplacian nonlinear diffusion equation rate of convergence 65m06 65m12 65m15 35r11 35k55