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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 Qk elements on cartesian meshes, in the sense that the time-step τ 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(L) 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(L) optimal error estimates for the pressure. Numerical experiments are given to validate our main results.

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