Summary: We investigate the temporal accuracy of two generalized-\(\alpha\) schemes for the incompressible Navier-Stokes equations. In a widely-adopted approach, the pressure is collocated at the time step \(t_{n+1}\) while the remainder of the Navier-Stokes equations is discretized following the generalized-\(\alpha\) scheme. That scheme has been claimed to be second-order accurate in time. We developed a suite of numerical code using inf-sup stable higher-order non-uniform rational B-spline (NURBS) elements for spatial discretization. In doing so, we are able to achieve high spatial accuracy and to investigate asymptotic temporal convergence behavior. Numerical evidence suggests that only first-order accuracy is achieved, at least for the pressure, in this aforesaid temporal discretization approach. On the other hand, evaluating the pressure at the intermediate time step \(t_{n+\alpha_f}\) recovers second-order accuracy, and the numerical implementation is simplified. We recommend this second approach as the generalized-\(\alpha\) scheme of choice when integrating the incompressible Navier-Stokes equations.
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