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

Abstract

We investigate the temporal accuracy of two generalized-α schemes for the incompressible Navier-Stokes equations. In a popular approach, the pressure is collocated at the time step t n+1 while the remainder of the Navier-Stokes equations is discretized following the rule of the generalized-α scheme. This scheme has been claimed to be secondorder accurate in time. 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 high spatial accuracy and to perform temporal refinement without introducing stabilization terms, which degenerate at small time steps. Numerical evidence suggests that only first-order accuracy is achieved, at least for the pressure, in this aforesaid temporal discretization approach. Evaluating the pressure at the intermediate time step recovers second-order accuracy, and the numerical implementation becomes simpler. The second approach is recommended as the generalized-α scheme of choice when integrating the incompressible Navier-Stokes equations.