Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow

The main purpose of this paper is to study the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with multi-step implicit-explicit (IMEX) time discretization schemes, for solving time-dependent incompressible fluid flows. We will give theoretical analysis for the Oseen equation, and assess the performance of the schemes for incompressible Navier-Stokes equations numerically. For the Oseen equation, using first order IMEX time discretization as an example, we show that the IMEX-LDG scheme is unconditionally stable for \mathcal {Q}_k elements on cartesian meshes, in the sense that the time-step \tau is only required to be bounded from above by a positive constant independent of the spatial mesh size h. Furthermore, by the aid of the Stokes projection and an elaborate energy analysis, we obtain the L^{\infty }(L^2) optimal error estimates for both the velocity and the stress (gradient of velocity), in both space and time. By the inf-sup argument, we also obtain the L^{\infty }(L^2) optimal error estimates for the pressure. Numerical experiments are given to validate our main results.