In this paper, we propose a unified numerical framework for the time-dependent incompressible Navier–Stokes equation which yields the $H^1$-, $H(\text{div})$-conforming, and discontinuous Galerkin methods with the use of different viscous stress tensors and penalty terms for pressure robustness. Under minimum assumption on Galerkin spaces, the semi- and fully-discrete stability is proved when a family of implicit Runge–Kutta methods are used for time discretization. Furthermore, we present a unified discussion on the penalty term. Numerical experiments are presented to compare our schemes with classical schemes in the literature in both unsteady and steady situations. It turns out that our scheme is competitive when applied to well-known benchmark problems such as Taylor–Green vortex, Kovasznay flow, potential flow, lid driven cavity flow, and the flow around a cylinder.

在本文中，我们为时间相关的不可压缩Navier–Stokes方程提出了一个统一的数值框架，该方程产生了 $H^{\mathrm{1个}}$-， $H（\text{div}）$-一致且不连续的Galerkin方法，使用了不同的粘性应力张量和压力稳健性的惩罚项。在Galerkin空间的最小假设下，当使用一系列隐式Runge-Kutta方法进行时间离散化时，证明了半离散和完全离散的稳定性。此外，我们对惩罚条款进行统一讨论。在非稳态和稳态下，都进行了数值实验，以比较我们的方案和文献中的经典方案。事实证明，当将其应用于著名的基准问题（例如泰勒-格林涡旋，Kovasznay流量，势能流量，盖驱动的腔体流量以及圆柱体流量）时，我们的方案具有竞争力。