A note on the accuracy of the generalized-$\alpha$ scheme for the incompressible Navier-Stokes equations

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.