Local discontinuous Galerkin method based on a family of second-order time approximation schemes for fractional mobile/immobile convection-diffusion equations

In this article, we introduce a local discontinuous Galerkin (LDG) method combined with the generalized second-order backward difference formula with a shifted parameter θ (BDF2- θ ) to solve the fractional mobile/immobile convection-diffusion equations, where the temporal direction is approximated by the generalized BDF2- θ and the spatial direction is discretized by the LDG method. We prove the stability for the numerical scheme and derive the rigorous error results that are related to the regularity of the solution. In order to investigate the correctness of the theoretical results and the effectiveness of the algorithm, we provide some numerical tests with Pk (k=1,3) elements for periodic boundary conditions and Pk (k=0,1,2,3) elements for compactly supported boundary conditions. Especially, with a comparison to the numerical scheme without the starting part, the corrected scheme yielded by adding the starting part can restore the second-order convergence rate for nonsmooth problems.