Abstract

For discontinuous Galerkin methods approximating convection diffusion equations, explicit time marching
is expensive since the time step is restricted by the square of the spatial mesh size. Implicit methods,
however, would require the solution of non-symmetric, non-positive definite and nonlinear systems, which
could be difficult. The high order accurate implicit-explicit (IMEX) Runge-Kutta or multi-step time
marching, which treats the diffusion term implicitly (often linear, resulting in a linear positive-definite
solver) and the convection term (often nonlinear) explicitly, can greatly improve computational efficiency.
We prove that certain IMEX time discretizations, up to third order accuracy, coupled with local discontinuous
Galerkin method for the diffusion term treated implicitly, and regular discontinuous Galerkin method
for the convection term treated explicitly, are unconditionally stable (the time step is upper bounded only
by a constant depending on the diffusion coefficient but not on the spatial mesh size) and optimally convergent.
The results also hold for drift-diffusion model in semiconductor device simulations, where a convection
diffusion equation is coupled with an electrical potential equation. Numerical experiments confirm the
good performance of such schemes. This is a joint work with Haijin Wang, Qiang Zhang and Yunxian Liu.