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

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.