Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations

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