Discontinuous Galerkin method for fractional convection-diffusion equations

TL;DR

This paper develops a discontinuous Galerkin method for solving fractional convection-diffusion equations involving fractional Laplacians of order between 1 and 2, providing stability and convergence analysis with numerical validation.

Contribution

It introduces a novel DG scheme for fractional convection-diffusion equations expressed as a system of differential and integral equations, with proven stability and optimal convergence rates.

Findings

Proves stability of the proposed method.

Establishes optimal convergence order O(h^{k+1}) for subdiffusion.

Demonstrates convergence order O(h^{k+1/2}) for general fractional convection-diffusion.

Abstract

We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a composite of first order derivatives and fractional integrals of order $2-\alpha$, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations. We prove stability and optimal order of convergence O($h^{k+1}$) for subdiffusion, and an order of convergence of ${\cal O}(h^{k+1/2})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.