SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2859-2886, December 2023.
Abstract. In this paper we consider the application of implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly. Both the second-order backward differentiation and the Crank–Nicolson methods are considered for time discretization, resulting in a scheme similar to Gear’s method on the one hand and to the Adams–Bashforth of second order on the other. 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 a high Reynolds number. Under suitable Courant conditions we prove stability of Gear’s scheme in this regime. The stabilization allows us to prove error estimates of order [math]. Here [math] is the mesh parameter, [math] is the polynomial order, and [math] the time step. Finally we show that for inviscid flow (or underresolved viscous flow) the IMEX scheme can be written as a fractional step method in which only a mass matrix is inverted for each velocity component and a Poisson-type equation is solved for the pressure.

中文翻译：

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高雷诺数下 Oseen 方程的隐式-显式时间离散化及其在分数阶方法中的应用

《SIAM 数值分析杂志》，第 61 卷，第 6 期，第 2859-2886 页，2023 年 12 月。
摘要。在本文中，我们考虑对不可压缩 Oseen 方程应用隐式-显式 (IMEX) 时间离散化。压力速度耦合和粘性项被隐式处理，而对流项被显式处理。二阶向后微分法和 Crank-Nicolson 方法都被考虑用于时间离散化，从而产生一方面类似于 Gear 方法，另一方面类似于二阶 Adams-Bashforth 的方案。对于空间中的离散化，我们考虑具有梯度跳跃稳定性的有限元方法。稳定项确保了等阶插值的 inf-sup 稳定性以及高雷诺数下的鲁棒性。在适当的 Courant 条件下，我们证明了 Gear 方案在此状态下的稳定性。稳定性使我们能够证明阶[数学]的误差估计。这里 [math] 是网格参数，[math] 是多项式阶数，[math] 是时间步长。最后，我们表明，对于无粘流（或未解析的粘性流），IMEX 方案可以写为分数阶方法，其中仅对每个速度分量求逆质量矩阵，并求解压力的泊松型方程。