Stability analysis and error estimates of implicit–explicit Runge–Kutta local discontinuous Galerkin methods for nonlinear fractional convection–diffusion problems

Abstract

In this paper, we shall present three fully discrete local discontinuous Galerkin (LDG) methods, coupled with implicit–explicit (IMEX) time discretization up to three order, for solving nonlinear fractional convection–diffusion problems with a fractional diffusion operator of order $\rho$ $(1<\rho <2)$ defined through the fractional Laplacian. In the time discretization, the convection term is treated explicitly and the fractional diffusion term implicitly. The fractional operator of order $\rho$ is expressed as a composite of first-order derivatives and a fractional integral of order $2-\rho$. We show that the IMEX-LDG schemes are unconditionally energy stable for nonlinear fractional convection–diffusion problems by the aid of energy analysis, in the sense that the time step $\tau$ is only required to be upper bounded by a constant which depends on the diffusion coefficient, but is independent of the mesh size h. We also obtain optimal error estimates in both space and time for the second- and third-order IMEX Runge–Kutta time-marching coupled with LDG spatial discretization, under the same temporal condition, if a monotone numerical flux is adopted for the convection. The analysis is confirmed by numerical examples.