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 α (1<α<2) defined through the fractional Laplacian. The fractional operator of order α is expressed as a composite of first order derivatives and a fractional integral of order 2-α. 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 O(hᵏ⁺¹) for the fractional diffusion problem, and an order of convergence of O(hk+1/2) is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.