论文标题
关于不可压缩的Navier-Stokes方程的通用$α$方案准确性的注释
A note on the accuracy of the generalized-$α$ scheme for the incompressible Navier-Stokes equations
论文作者
论文摘要
我们研究了不可压缩的Navier-Stokes方程的两个广义 - $α$方案的时间准确性。常规方法通过向后的Euler方法处理压力，同时使用广义-YAM $α$方法离散其余的Navier-Stokes方程。我们使用INF-SUP稳定的高阶非均匀有理B-Spline（NURB）元素开发了一套数值代码，以进行空间离散化。这样一来，我们就能达到非常高的空间精度，此外，在不考虑稳定项的情况下进行时间精致，可以在很小的时间步长下退化。数值实验表明，在这种上述方法中，至少对于压力，仅实现了一阶精度。评估中间时间步骤的压力恢复二阶精度，实际上，数值实现变得更加简单。因此，尽管可以将压力视为执行不可压缩限制的拉格朗日乘数，但其时间离散化并非独立，应受到广义 - $α$方法的约束，以维持整体算法的二阶准确性。
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.
