On numerical methods and error estimates for degenerate fractional convection-diffusion equations

TL;DR

This paper introduces a convergent numerical method for nonlinear nonlocal degenerate convection-diffusion equations with fractional derivatives, and develops a Kuznetsov type theory to obtain optimal error estimates for these methods.

Contribution

It presents a new numerical scheme and a theoretical error analysis for a broad class of fractional and degenerate equations, including Levy diffusion and porous medium types.

Findings

Numerical method converges for nonlinear nonlocal equations.

Error estimates are optimal even for fractional derivatives between 1 and 2.

Applicable to equations with Levy diffusion and degenerate cases.

Abstract

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods - even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations.