Second-order finite-element projection method for 3D flows J.-L. Guermond I and L. Quartapelle 2 1 LIMSI-CNRS, BP 133, 91403 Orsay, France, (guermond@limsi.fr) 2 Dipartimento di Fisica del Politecnico di Milano, Piazza L. da Vinci, 32 20133 Milano, Italy. Key Words: INCOMPRESSIBLE NAVIER-STOKES EQUATIONS, INCREMENTAL PROJECTION METHOD, SECOND ORDER SCHEME, FINITE ELEMENTS 1 Introduction Achieving high-order time accuracy in the solution of the incompressible Navier-Stokes equations by means of projection methods is a nontrivial task. In fact, a basic feature of projection methods is the uncoupling of the advection-diffusion mechanism from the incompressibility condition, the con- sequence of this uncoupling being the introduction of a time-sp]itting error that is an obstacle to develop high order schemes. In particular, the accu- racy of the nonincremental projection method introduced by Chorin (1969) and Temam (1968) is limited by an irreducible time-splitting error of O(At) Rannacher (1992) that makes second-order accuracy impossible. This limi- tation is not present in the incremental version of the projection method, also known as "pressure correction method", originally proposed by Goda (1979) in a finite difference context. The finite element counterpart of this scheme based on a first-order Euler time-discretization has been analyzed in Guermond (1994) and employed successfully in Guermond and Quartapelle (1997), where its time-splitting error has been numerically shown to be of O((At)2). This result has been proved in Guermond (1997), where a new (At)2-accurate projection scheme based on the three-level Backward Differ- ence Formula has been proposed. The aim of this paper is to demonstrate the second-order accuracy of the incremental BDF method by means of nu- merical tests and to illustrate a finite element implementation of this scheme to simulate three dimensional realistic flows. 2 Incremental projection method Consider the unsteady Navier-Stokes problem: Find the velocity u and the pressure p (up to a constant) so that, Ult=0 = u0 and ou _ ~v2u + (u .v)u + vp = $, v. u = 0, (2.1) ulo~ = b, where ~ is the viscosity, f is a known body force, b is the velocity prescribed on the boundary 0~, and u0 is the divergence-free initial velocity field. The