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

Abstract

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.

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• The nested block preconditioning technique for the incompressible Navier-Stokes equations with emphasis on hemodynamic simulations
• A reduced unified continuum formulation for vascular fluid-structure interaction
• Generalized-alpha scheme in the PFEM for velocity-pressure and displacement-pressure formulations of the incompressible Navier-Stokes equations
• On the Incorporation of Obstacles in a Fluid Flow Problem Using a Navier-Stokes-Brinkman Penalization Approach
• An open-source computational framework for immersed fluid-structure interaction modeling using FEBio and MFEM