Quelques résultats nouveaux sur les méthodes de projection

doi:10.1016/S0764-4442(01)02157-7

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics

Volume 333, Issue 12, 15 December 2001, Pages 1111-1116

Comptes Rendus de l'...

https://doi.org/10.1016/S0764-4442(01)02157-7 Get rights and content

Résumé

Nous revisitons les méthodes de projection de Chorin–Temam du type correction de pression et nous introduisons une nouvelle classe de méthodes que nous appelons correction de vitesse. Pour une variante de la méthode de correction de pression et une variante de la méthode de correction de vitesse, nous prouvons une convergence en $O (δt^{2})$ sur la vitesse en norme L² et une convergence en $O (δt^{3/2})$ sur la vitesse en norme H¹ et la pression en norme L². Nous montrons aussi que les méthodes de correction de vitesse fournissent le bon cadre fonctionel pour l'analyse des méthodes de splitting introduites dans [4,3]. Cette Note fournit donc en corollaire le premier résultat de stabilité et de convergence pour les méthodes [4,3].

Abstract

We revisit fractional step projection methods for solving the Navier–Stokes equations. We study a variant of pressure-correction methods and introduce a new class of velocity-correction methods. We prove stability and $O (δt^{2})$ convergence in the L² norm of the velocity for both variants. We also prove $O (δt^{3/2})$ convergence in the H¹ norm of the velocity and the L² norm of the pressure. We show that the new family of projection methods can be related to a set of methods introduced in [4,3]. As a result, this Note provides the first rigorous proof of stability and convergence of the methods introduced in [4,3].

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Cited by (17)

A fractional-step DG-FE method for the time-dependent generalized Boussinesq equations
2023, Communications in Nonlinear Science and Numerical Simulation
Citation Excerpt :
The fractional-step method is based on the idea of totally decoupling the nonlinearity from incompressibility in Navier–Stokes equations. Also known as the projection method, this method was first introduced by Chorin [28] and Temam [29], and developed into various kinds of schemes (see [30–40]). The DG method was first introduced by Reed and Hill for hyperbolic equations [41], and analysed by Lesaint and Raviart in [42,43].
In this work a fractional-step DG-FE method for the time-dependent generalized Boussinesq equations is proposed and analysed. The scheme is composed of two steps. In the first step the original problem is reduced into several scalar elliptic equations. An intermediate velocity and temperature are solved simultaneously. Then in the second step, the incompressibility constraint is enforced and velocity is corrected to be discretely divergence free. Moreover, the introduced elliptic term in the correction step enables the imposition of correct Dirichlet boundary conditions at each temporal step, avoiding the artificial boundary layer introduced by classical pressure correction method. DG-FE discretization strategy is utilized, in which the discontinuous Galerkin spacial discretization for flow equations is employed to obtain local mass conservation and traditional finite element spacial discretization is adopted for heat equation to reduce degrees of freedom. By choosing different symmetry and penalty parameters, SIPG-FE and NIPG-FE methods can be utilized. The consistency and stability of both methods are proved. Preliminary error estimates proving the optimal spacial order and suboptimal temporal order are carried out. Based on a different error equation and the preliminary error estimates, the optimal temporal convergence order is obtained. Numerical tests including a benchmark simulating square cavity flow are then presented, to verify the theoretical analysis and validate the method.
A rotational velocity-correction projection method for unsteady incompressible magnetohydrodynamics equations
2020, Computers and Mathematics with Applications
Citation Excerpt :
In [19], a decoupling penalty finite element method for the stationary incompressible magnetohydrodynamics equation was shown. As we know, in the literature, some fractional step methods are often referred as projection methods, which can be classified into three classes, namely, the pressure-correction methods (see [20,21]), the velocity-correction methods (see [22,23]), and the consistent splitting methods (see [24,25]). Pressure-correction schemes are time-marching techniques composed of two sub-steps for each time step: the pressure is treated explicitly or ignored in the first sub-step and is corrected in the second.
This paper provides a rotational velocity-correction projection method for the unsteady incompressible magnetohydrodynamics (MHD) equations. In this method, the electrical field variable $E$ was eliminated. In this way, the Gauss’s law can be conserved. The velocity variable $u$ and the pressure field $p$ are solved by a rotational velocity-correction projection method. The optimal error estimates of the $L^{2}$ -norm will be given. Finally, we shall present some numerical results to confirm our analysis. It shows that our method is effective and can conserve the Gauss’s Law very well.
Optimal first-order error estimates of a fully segregated scheme for the Navier–Stokes equations
2017, Journal of Computational and Applied Mathematics
Citation Excerpt :
Some current variants of projection methods are: rotational pressure-correction schemes [4–6], velocity-correction schemes [7,8], consistent-splitting schemes [9,6,10] and penalty pressure-projection schemes [11–13].
A first-order linear fully discrete scheme is studied for the incompressible time-dependent Navier–Stokes equations in three-dimensional domains. This scheme is based on an incremental pressure projection method and decouples each component of the velocity and the pressure, solving in each time step, a linear convection–diffusion problem for each component of the velocity and a Poisson–Neumann problem for the pressure.
Using an inf–sup stable and continuous finite-elements approach of order $O (h)$ in space, unconditional optimal error estimates of order $O (k + h)$ are deduced for velocity and pressure (without imposing constraints on the mesh size $h$ and the time step $k$ ).
Finally, some numerical results are performed to validate the theoretical analysis, and also to compare the studied scheme with other current first-order segregated schemes.
A robust iterative scheme for finite volume discretization of diffusive flux on highly skewed meshes
2009, Journal of Computational and Applied Mathematics
Citation Excerpt :
It could also be used directly in algorithms where a Poisson equation is inevitable, like projection velocity–pressure coupling algorithms [15–17].
A new approach for diffusive flux discretization on a nonorthogonal mesh for finite volume method is proposed. This approach is based on an iterative method, Deferred correction introduced by M. Peric [J.H. Fergizer, M. Peric, Computational Methods for Fluid Dynamics, Springer, 2002]. It converges on highly skewed meshes where the former approach diverges. A convergence proof of our method is given on arbitrary quadrilateral control volumes. This proof is founded on the analysis of the spectral radius of the iteration matrix. This new approach is applied successfully to the solution of a Poisson equation in quadrangular domains, meshed with highly skewed control volumes. The precision order of used schemes is not affected by increasing skewness of the grid. Some numerical tests are performed to show the accuracy of the new approach.
An overview of projection methods for incompressible flows
2006, Computer Methods in Applied Mechanics and Engineering
Citation Excerpt :
Thus, if (2.6) has a steady-state solution as t → +∞ (assuming f is independent of t), and if the discrete steady-state problem (8.2) has spurious pressure modes, then all the algorithms (3.17)–(3.19), (4.8)–(4.10), or (5.9)–(5.11) will give at steady-state a pressure field that is defined up to an arbitrary spurious mode. The above argument is developed in more details in Guermond and Quartapelle [22], Guermond and Shen [26], and Minev [38]. In a finite element method, the Neumann boundary condition is usually enforced naturally.
A series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper. These methods are often referred to in the literature as projection methods, and can be classified into three classes, namely the pressure-correction methods, the velocity-correction methods, and the consistent splitting methods. For each class of schemes, theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed. The essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners.
New approach for the discretization of diffusive flux on a orthogonal mesh
2006, Comptes Rendus - Mecanique
Nous proposons une nouvelle approche pour la discrétisation des flux diffusifs sur un maillage non orthogonal basé sur la ‘Deferred correction’ introduite par Peric. Cette nouvelle méthode est appliquée avec succès à la résolution du problème de Poisson dans un domaine en forme de parallélogramme maillé avec des volumes de contrôle à obliquité sévère sans que l'ordre de précision des schémas mis en œuvre ne soit dégradé. L'étude comparative de la valeur du rayon spectral des matrices d'itération nous a permis d'expliquer pourquoi notre approche de Deferred correction est meilleure que l'ancienne approche de Peric, qui diverge en présence de forte obliquité. Pour citer cet article : Y.M. Ahipo, Ph. Traore, C. R. Mecanique 334 (2006).
We propose a new approach for the discretization of diffusive flux on a non orthogonal mesh based on the Deferred correction introduced by Peric. This new method is applied successfully to the solution of a Poisson problem in quadrangular domains meshed with very distorted control volumes. The interest of this approach lies in the fact that the precision of the used schemes is conserved despite meshes distortion level. The comparative study of the value of the spectral radius of iteration matrix enables us to explain why our Deferred correction approach is better than that of Peric, which diverges on highly skewed meshes. To cite this article: Y.M. Ahipo, Ph. Traore, C. R. Mecanique 334 (2006).

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Quelques résultats nouveaux sur les méthodes de projectionNew results on several projection methods

Résumé

Abstract

J. Comput. Phys.

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

Modél. Math. Anal. Numér. (M2AN)

High-order splitting methods for the incompressible Navier–Stokes equations

J. Comput. Phys.