Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

Abstract:

In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the $L^2$ norm of the numerical solution does not increase in time, under the time step condition $\tau \le \mathcal {F}(h/c, d/c^2)$, with the convection coefficient $c$, the diffusion coefficient $d$, and the mesh size $h$. The function $\mathcal {F}$ depends on the specific IMEX temporal method, the polynomial degree $k$ of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes $\tau \lesssim h/c$ in the convection-dominated regime and it becomes $\tau \lesssim d/c^2$ in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.