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-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-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+ f recovers second-order accuracy, and the numerical implementation is simplified. We recommend this second approach as the generalized-scheme of choice when integrating the incompressible Navier-Stokes equations.

A reduced unified continuum formulation for vascular fluid-structure interaction

“…Evaluating pressure at t n+α f recovers second-order accuracy for the overall algorithm, simplifies the implementation, and resolves a troubling issue in geometric multiscale modeling. Interested readers are referred to [49] for details.…”

Numerical investigation of abdominal aortic aneurysm hemodynamics using the reduced unified continuum formulation for vascular fluid-structure interaction

“…We additionally emphasize novel aspects of our numerical strategy pertaining to the temporal discretization and linear solver. Temporal discretization of the entire FSI system is performed with the generalized-α scheme, in which velocity and pressure are uniformly evaluated at the intermediate time step to achieve uniform second-order temporal accuracy, in direct contrast to the predominant dichotomous approach offering only first-order accuracy of pressure [4]. Furthermore, block preconditioning of a monolithically coupled FSI system is made possible for the first time through a segregated predictor multi-corrector algorithm preserving the block structure of the incompressible Navier-Stokes equations in the implicit solver's fully consistent linear system.…”

“…The generalized-α method is applied for temporal discretization of the semi-discrete FSI formulation derived in Section 2.2, in which both velocity and pressure are collocated at the intermediate time step to achieve uniform second-order temporal accuracy [4]. Without loss of consistency, the fully discrete scheme is solved with a segregated predictor multi-corrector algorithm preserving the two-by-two block structure of the incompressible Navier-Stokes equations in the implicit solver's associated linear system.…”

“…Furthermore, uniform temporal discretization of the entire FSI system is achieved via the generalized-α scheme, in which velocity and pressure are concurrently collocated at the intermediate time step for second-order temporal accuracy. This is in contrast to the dichotomous approach commonly adopted in the computational fluid dynamics (CFD) and FSI communities, which we recently found to yield only first-order temporal accuracy for pressure [4]. Without loss of consistency, a segregated predictor multi-corrector algorithm is designed to preserve the same block structure as for the incompressible Navier-Stokes equations in the implicit solver's associated linear system.…”

A Divergence-Conforming Hybridized Discontinuous Galerkin Method for the Incompressible Magnetohydrodynamics Equations

“…It should be noted that conventionally the pressure field is evaluated at the (n + 1) st time step rather than the (n + α f ) th time step. However, as recently shown in [38], this limits the accuracy of the pressure field to first-order-in-time. Moreover, we have found that evaluating the trace fields at the (n + 1) st time step rather than the (n + α f ) th time step limits the accuracy of all fields to first-order-in-time.…”