Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit–Explicit Backward Difference Formulas up to Fifth Order for Convection–Diffusion Equation

In this paper, the stability analysis and optimal error estimates are presented for a kind of fully discrete schemes for solving one-dimensional linear convection–diffusion equation with periodic boundary conditions. The fully discrete schemes are defined with local discontinuous Galerkin (LDG) spatial discretization methods coupled with implicit–explicit (IMEX) temporal discretization methods based on backward difference formulas (BDF). By combining an improved multiplier technique used in the stability analysis for multistep methods and the technique to deal with derivative and jump in LDG methods, we establish a general framework of stability analysis for the corresponding fully discrete LDG–IMEX–BDF schemes up to fifth order in time. The considered schemes are proved to be unconditionally stable, in the sense that a properly defined “discrete energy” is dissipative if the time step is upper bounded by a constant which is independent of the mesh size. Optimal orders of the \(L^2\) norm accuracy in both space and time are also proved by energy analysis. Numerical tests are presented to validate the theoretical results.