An analysis and comparison of the time accuracy of fractional‐step methods for the Navier–Stokes equations on staggered grids

Abstract

SUMMARYFractional-step methods solve the unsteady Navier-Stokes equations in a segregated manner, and can be implemented with only a single solution of the momentum=pressure equations being obtained at each time step, or with the momentum=pressure system being iterated until a convergence criterion is attained. The time accuracy of such methods can be determined by the accuracy of the momentum=pressure coupling, irrespective of the accuracy to which the momentum equations are solved. It is shown that the time accuracy of the basic projection method is ÿrst-order as a result of the momentum=pressure coupling, but that by modifying the coupling directly, or by modifying the intermediate velocity boundary conditions, it is possible to recover second-order behaviour. It is also shown that pressure correction methods, implemented in non-iterative or iterative form and without special boundary conditions, are second-order in time, and that a form of the non-iterative pressure correction method is the most e cient for the problems considered.

A nonhydrostatic version of FVCOM: 1. Validation experiments

“…Then a fractional step method [Chorin, 1968] was used, through which the hydrostatic pressure is computed explicitly from the free surface elevation and the density field, and the nonhydrostatic pressure is determined on the condition that the local velocity field must be divergence-free. We have implemented both the "projection" and "pressure correction" fractional step methods [Armfield and Street, 2002] into FVCOM. In the projection method, the momentum equations are first integrated using a hydrostatic pressure gradient to obtain the intermediate velocities.…”

An investigation of the behaviour of planar jets using a time accurate fractional step scheme

“…The discrete advection operator, the discrete gradient, the discrete Laplace operator and the discrete divergence are represented by H, G, L and D respectively [1].…”

“…Armfield and Street have compared this method and analysed several other fractional step methods on staggered and non-staggered grids, their findings demonstrate the ad-C557 vantages of the pressure correction approach in terms of increased efficiency whilst retaining overall second order accuracy [1,2].…”

Preconditioning in parallel for fractional step Navier--Stokes solvers

“…This results in two separate equations, one involving momentum terms solved for the velocity and the other for the pressure. Various forms of fractionalstep schemes have been developed to achieve better efficiency and time accuracy; Armfield and Street [6] overviewed and compared some schemes. Application of the fractional-step Navier-Stokes method with second order pressure correction (p2) and negligible body force, using Adams-Bashforth and Crank-Nicolson methods for the time discretisation of the advective and C22 diffusive terms respectively, results in the equations…”