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pre_cor - MATHEMATICS OF COMPUTATION Volume 73 Number 248...

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MATHEMATICS OF COMPUTATION Volume 73, Number 248, Pages 1719–1737 S 0025-5718(03)01621-1 Article electronically published on December 19, 2003 ON THE ERROR ESTIMATES FOR THE ROTATIONAL PRESSURE-CORRECTION PROJECTION METHODS J. L. GUERMOND AND JIE SHEN Abstract. In this paper we study the rotational form of the pressure-correc- 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 Navier-Stokes 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 predictor-corrector 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 pressure-correction methods (see [6, 20]) and splitting techniques based on extrapolated pressure boundary con- ditions (see [10, 9]). Pressure-correction 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 second-order error estimates for the velocity have been proved by E and Liu [4] and Shen [15] in the semi-discrete 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 pressure-correction 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. Navier-Stokes equations, projection methods, fractional step methods, incompressibility, finite elements, spectral approximations. The work of the second author is partially supported by NFS grants DMS-0074283 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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1720 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 pressure-correction scheme, which we shall hereafter refer to as the rotational form
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