Un résultat de convergence d'ordre deux en temps pour l'approximation des équations de Navier–Stokes par une technique de projection incrémentale

Abstract

Abstract. The Navier-Stokes equations are approximated by means of a fractional step, ChorinTemam projection method; the time derivative is approximated by a three-level backward finite difference, whereas the approximation in space is performed by a Galerkin technique. It is shown that the proposed scheme yields an error of O(δt 2 + h l+1 ) for the velocity in the norm of l 2 (L 2 (Ω) d ), where l ≥ 1 is the polynomial degree of the velocity approximation. It is also shown that the splitting error of projection schemes based on the incremental pressure correction is of O(δt 2 ) independent of the approximation order of the velocity time derivative.Résumé. Leséquations de Navier-Stokes sont approchées en temps par une schéma de différentiation rétrograde d'ordre deux et une technique de pas fractionnaire du type projection de Chorin-Temam ; l'approximation spatiale est réalisée par une technique de Galerkin. On montre qu'en temps fini, le schéma est d'ordre O(δt 2 + h l+1 ) pour la vitesse dans la norme l 2 (L 2 (Ω) d ), où l ≥ 1 est le degré polynomial d'approximation de la vitesse. On montre aussi que l'erreur de fractionnement des schémas de projection basés sur une correction de pression incrémentale est d'ordre O(δt 2 ), que l'approximation de la dérivée temporelle de la vitesse soit d'ordre un ou d'ordre deux.

On the approximation of the unsteady Navier-Stokes equations by finite element projection methods

“…They yield first order accuracy in time in the natural norms. The possibility of second-order accurate projection schemes in the presence of solid no-slip boundaries has not been considered in the present work but second order accuracy in time is possible and proof of convergence are reported in Guermond [18].…”

“…Most of what is said in this paper applies also to the original nonincremental projection algorithm. In short, both algorithms converge, but the incremental one has a better rate of convergence than the nonincremental one in the natural norms (see Guermond [16], [18], and Rannacher [26]). …”

Efficient and scalable discretization of the Navier–Stokes equations with LPS modeling

“…In this way, after applying a standard incremental pressure-correction approach (cf. [26,41] for derivation in the case of equal-order and inf-sup stable FE, respectively) to system (3.2), the fully discrete semiimplicit formulation consists in solving, for n = 0, . .…”

Modeling and Numerical Approximation of Two-Phase Incompressible Flows by a Phase-Field Approach

“…The most effective way to decouple the computation of pressure from that of the velocity is to employ a projection type scheme which was originally proposed by Chorin [12] and Temam [69]. Many improved versions have been introduced over the last forty years (cf., for instance, [26,41,72,52,40,66,54,67,70,20,30,31,33,39]); and an extensive literature has been devoted to the numerical analysis of these projection type schemes (cf., for instance, [59,66,36,18,19,67,55,27,58,6,30,31,33,51,49]). We refer to the recent review paper [32] for a detailed account on the various type of projection schemes.…”

“…The only complication is that the term (3ũ k+1 − 4u k + u k−1 ,ũ k+1 ) needs to be treated carefully, we refer to [27,33] for more detail on this matter. As for the error analysis, it is shown (cf.…”