Implicit-Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection-Diffusion Problems

Local discontinuous Galerkin methods with generalized alternating numerical fluxes coupled with implicit-explicit time marching for solving convection-diffusion problems is analyzed in this paper, where the explicit part is treated by a strong-stability-preserving Runge-Kutta scheme, and the implicit part is treated by an L-stable diagonally implicit Runge-Kutta method. Based on the generalized alternating numerical flux, we establish the important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient, which plays a key role in obtaining the unconditional stability of the proposed schemes. Also by the aid of the generalized Gauss-Radau projection, optimal error estimates can be shown. Numerical experiments are given to verify the stability and accuracy of the proposed schemes with different numerical fluxes.