MATHEMATICS OF COMPUTATION
Volume 73, Number 248, Pages 1719–1737
S 00255718(03)016211
Article electronically published on December 19, 2003
ON THE ERROR ESTIMATES
FOR THE ROTATIONAL PRESSURECORRECTION
PROJECTION METHODS
J. L. GUERMOND AND JIE SHEN
Abstract.
In this paper we study the rotational form of the pressurecorrec
tion method that was proposed by Timmermans, Minev, and Van De Vosse.
We show that the rotational form of the algorithm provides better accuracy in
terms of the
H
1
norm of the velocity and of the
L
2
norm of the pressure than
the standard form.
1.
Introduction
There are numerous way to discretize the unsteady incompressible NavierStokes
equations in time. Undoubtedly, the most popular one consists of using projection
methods. This class of techniques has been introduced by Chorin and Temam
[2, 3, 17]. They are time marching algorithms based on a fractional step technique
that may be viewed as a predictorcorrector strategy aiming at uncoupling viscous
diﬀusion and incompressibility eﬀects. The method proposed originally, although
simple, is not satisfactory since its convergence rate is irreducibly limited to
O
(
δt
).
This limitation comes from the fact that the method is basically an artiﬁcial com
pressibility technique as shown in [11] and [13]. To cure these problems, numerous
modiﬁcations have been proposed, among which are pressurecorrection methods
(see [6, 20]) and splitting techniques based on extrapolated pressure boundary con
ditions (see [10, 9]).
Pressurecorrection methods are widely used and have been extensively ana
lyzed. These schemes are composed of two substeps by time step: the pressure is
made explicit in the ﬁrst substep and is corrected in the second one by projecting
the provisional velocity onto the space of incompressible vector ﬁelds. Rigorous
secondorder error estimates for the velocity have been proved by E and Liu [4] and
Shen [15] in the semidiscrete case and by Guermond [7] and E and Liu [5] in the
fully discrete case. We refer also to [16] and [1] for diﬀerent proofs based on normal
mode analysis in the half plane and in a periodic channel, respectively.
It is well known that standard pressurecorrection schemes still suﬀer from the
nonphysical pressure boundary condition which induces a numerical boundary layer
Received by the editor February 11, 2002 and, in revised form, March 2, 2003.
2000
Mathematics Subject Classiﬁcation.
Primary 65M12, 35Q30, 76D05.
Key words and phrases.
NavierStokes equations, projection methods, fractional step methods,
incompressibility, ﬁnite elements, spectral approximations.
The work of the second author is partially supported by NFS grants DMS0074283 and DMS
0311915. Part of the work was completed while this author was a CNRS “Poste Rouge” visitor
at LIMSI.
c
±
2003 American Mathematical Society
1719
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J. L. GUERMOND AND JIE SHEN
and, consequently, degrades the accuracy of the pressure approximation. In 1996,
Timmermans, Minev and Van De Vosse [19] proposed a modiﬁed version of the
pressurecorrection scheme, which we shall hereafter refer to as the rotational form
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 Spring '09
 Math, Numerical Analysis, Dirichlet boundary condition, J. L. GUERMOND, JIE SHEN, ROTATIONAL PRESSURECORRECTION METHOD

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