Third order implicit-explicit Runge-Kutta local discontinuous Galerkin methods with suitable boundary treatment for convection-diffusion problems with Dirichlet boundary conditions

To avoid the order reduction when third order implicit-explicit Runge-Kutta time discretization is used together with the local discontinuous Galerkin (LDG) spatial discretization, for solving convection-diffusion problems with time-dependent Dirichlet boundary conditions, we propose a strategy of boundary treatment at each intermediate stage in this paper. The proposed strategy can achieve optimal order of accuracy by numerical verification. Also by suitably setting numerical flux on the boundary in the LDG methods, and by establishing an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient and the given boundary conditions, we build up the unconditional stability of the corresponding scheme, in the sense that the time step is only required to be upper bounded by a suitable positive constant, which is independent of the mesh size. keywords. local discontinuous Galerkin method, implicit-explicit time discretization, convection-diffusion equation, Dirichlet boundary condition, order reduction. AMS. 65M12, 65M15, 65M60