A note on the accuracy of the generalized-α scheme for the incompressible Navier-Stokes equations

We investigate the temporal accuracy of two generalized- (Formula presented.) schemes for the incompressible Navier-Stokes equations. In a widely-adopted approach, the pressure is collocated at the time step t while the remainder of the Navier-Stokes equations is discretized following the generalized- (Formula presented.) 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 (Formula presented.) recovers second-order accuracy, and the numerical implementation is simplified. We recommend this second approach as the generalized- (Formula presented.) scheme of choice when integrating the incompressible Navier-Stokes equations.