Analysis of a local discontinuous Galerkin method for time‐fractional advection‐diffusion equations

Abstract

PurposeThe purpose of this paper is to develop a fully discrete local discontinuous Galerkin (LDG) finite element method for solving a time‐fractional advection‐diffusion equation.Design/methodology/approachThe method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space.FindingsBy choosing the numerical fluxes carefully the authors' scheme is proved to be unconditionally stable and gets L2 error estimates of O(hk+1+(Δt)2+(Δt)α/2hk+(1/2)). Finally Numerical examples are performed to illustrate the effectiveness and the accuracy of the method.Originality/valueThe proposed method is different from the traditional LDG method, which discretes an equation in spatial direction and couples an ordinary differential equation (ODE) solver, such as Runger‐Kutta method. This fully discrete scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. Numerical examples prove that the authors' method is very effective. The present paper is the authors' first step towards an effective approach based on the discontinuous Galerkin method for the solution of fractional‐order problems.