关于不可压Navier-斯托克斯方程的广义- $α$格式精度的注记
A note on the accuracy of the generalized-$α$ scheme for the incompressible Navier-Stokes equations
摘要：
研究了不可压Navier-斯托克斯方程的两种广义$α$格式的时间精度.传统的方法用向后欧拉法处理压力，而用广义$α$方法离散Navier-斯托克斯方程的其余部分。我们开发了一套数值代码使用inf-sup稳定的高阶非均匀有理B样条（NURBS）元素的空间离散。在这样做时，我们能够实现非常高的空间精度，而且，执行时间细化，而不考虑的稳定项，这可能会退化为小的时间步长。数值实验表明，在上述方法中，至少对于压力，仅实现一阶精度。在中间时间步的压力评估恢复二阶精度，和数值实现，事实上，变得更简单。因此，虽然压力可以被看作是一个拉格朗日乘子强制执行不可压缩性约束，其时间离散化是不独立的，应该受到广义$α$方法，以保持二阶精度的整体算法。
英文摘要：
We investigate the temporal accuracy of two generalized-$α$ 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-$α$ 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-$α$ method in order to maintain second-order accuracy of the overall algorithm.
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