Structure-preserving scheme for fractional nonlinear diffusion equations

TL;DR

This paper develops a numerical scheme for fractional nonlinear diffusion equations that preserves key properties like decay and extinction, supported by rigorous analysis and extensive simulations.

Contribution

It introduces a structure-preserving numerical scheme for fractional nonlinear diffusion equations, including a novel method for computing extinction times.

Findings

The scheme preserves algebraic decay in fractional porous medium equations.

It accurately captures the extinction phenomenon in fractional fast diffusion.

Numerical simulations confirm the convergence to asymptotic profiles near extinction.

Abstract

In this paper, we introduce and analyze a numerical scheme for solving the Cauchy-Dirichlet problem associated with fractional nonlinear diffusion equations. These equations generalize the porous medium equation and the fast diffusion equation by incorporating a fractional diffusion term. We provide a rigorous analysis showing that the discretization preserves main properties of the continuous equations, including algebraic decay in the fractional porous medium case and the extinction phenomenon in the fractional fast diffusion case. The study is supported by extensive numerical simulations. In addition, we propose a novel method for accurately computing the extinction time for the fractional fast diffusion equation and illustrate numerically the convergence of rescaled solutions towards asymptotic profiles near the extinction time.