We investigate the temporal accuracy of two generalized-$\alpha$ schemes for the incompressible Navier-Stokes equations. The conventional approach treats the pressure with the backward Euler method while discretizing the remainder of the Navier-Stokes equations with the generalized-$\alpha$ method. We developed a suite of numerical codes using inf-sup stable higher-order non-uniform rational B-spline (NURBS) elements for spatial discretization. In doing so, we are able to achieve very high spatial accuracy and, furthermore, to perform temporal refinement without consideration of the stabilization terms, which can degenerate for small time steps. Numerical experiments suggest that only first-order accuracy is achieved, at least for the pressure, in this aforesaid approach. Evaluating the pressure at the intermediate time step recovers second-order accuracy, and the numerical implementation, in fact, becomes simpler. Therefore, although the pressure can be viewed as a Lagrange multiplier enforcing the incompressibility constraint, its temporal discretization is not independent and should be subject to the generalized-$\alpha$ method in order to maintain second-order accuracy of the overall algorithm.

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关于不可压缩 Navier-Stokes 方程的广义 α 格式的准确性的说明

我们研究了不可压缩 Navier-Stokes 方程的两个广义 $\alpha$ 方案的时间精度。传统方法使用后向欧拉方法处理压力，同时使用广义-$\alpha$ 方法离散 Navier-Stokes 方程的其余部分。我们使用 inf-sup 稳定的高阶非均匀有理 B 样条 (NURBS) 元素开发了一套用于空间离散化的数值代码。这样做，我们能够实现非常高的空间精度，此外，在不考虑稳定项的情况下进行时间细化，稳定项可能会退化为小时间步长。数值实验表明，在上述方法中，至少对于压力而言，只能实现一阶精度。在中间时间步评估压力可以恢复二阶精度，而且数值实现实际上变得更简单。因此，虽然压力可以被视为强制不可压缩约束的拉格朗日乘数，但它的时间离散化不是独立的，应该服从广义-$\alpha$ 方法，以保持整个算法的二阶精度。