On numerical methods and error estimates for degenerate fractional convection-diffusion equations
This work provides a unified numerical framework and error analysis for a wide range of degenerate fractional equations, benefiting researchers in numerical analysis and applied mathematics.
The paper introduces a convergent numerical method for a broad class of nonlinear nonlocal degenerate convection-diffusion equations and develops a new Kuznetsov-type theory to obtain general, possibly optimal error estimates for fractional orders between 1 and 2.
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.