In this paper we continue the work on implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations that we started in \cite{BGG23} (E. Burman, D. Garg, J. Guzm\`an, {\emph{Implicit-explicit time discretization for Oseen's equation at high Reynolds number with application to fractional step methods}}, SIAM J. Numer. Anal., 61, 2859--2886, 2023). The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein we focus on the implicit-explicit Crank-Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions we prove stability of the implicit-explicit Crank-Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order $O(h^{k+\frac12} + \tau^2)$. Here $h$ is the mesh parameter, $k$ the polynomial order and $\tau$ the time step. Finally we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier-Stokes' equations.

翻译：本文延续了我们在\cite{BGG23}（E. Burman, D. Garg, J. Guzmàn, 《高雷诺数下Oseen方程的时间隐式-显式离散及其在分数步方法中的应用》，SIAM J. Numer. Anal., 61, 2859–2886, 2023）中关于不可压缩Oseen方程隐式-显式（IMEX）时间离散化的工作。其中，压力-速度耦合和粘性项采用隐式处理，而对流项则通过外推显式处理。本文重点研究了用于时间离散化的隐式-显式Crank-Nicolson方法。在空间离散化方面，我们采用基于梯度跳跃稳定化的有限元方法。该稳定项确保了等阶插值时的inf-sup稳定性以及在高雷诺数下的鲁棒性。在适当的库朗条件下，我们证明了该隐式-显式Crank-Nicolson格式在此情形下的稳定性。稳定化措施使我们能够证明阶数为$O(h^{k+\frac12} + \tau^2)$的误差估计，其中$h$为网格参数，$k$为多项式阶数，$\tau$为时间步长。最后，我们讨论了由IMEX格式推导出的几种分数步方法，并报告了将这些方法应用于Navier-Stokes方程时的数值算例对比结果。