Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The full numerical simulation of turbulent flows in the context of industrial applications remains a very challenging problem. One of the difficulties is the presence of a large number of interacting scales of different orders of magnitude, ranging from the macroscopic scale to the Kolmogorov dissipation scale. In order to better understand and simulate these interactions, new algorithms of incremental type have recently been introduced, stemming from the theory of infinite-dimensional dynamical systems (see, e.g., the algorithms of Foias, Jolly et al. (1988), Foias, Manley and Temam (1988), Laminie et al. (1993, 1994), Marion and Temam (1989, 1990), and Temam (1990)). These algorithms are based on decompositions of the unknown functions into a large- and a small-scale component, one of the underlying ideas being to approximate the corresponding attractor. In the context of finite element methods, the decomposition of a solution into small- and large-scale components appears when we consider hierarchical bases (Yserentant, 1986; Zienkiewicz et al., 1982). In the present article we derive several estimates of the size of the structures for the linear and nonlinear terms which correspond to interactions of different hierarchical components of the velocity field, and also their time variation. The one-step time variation of these terms can be smaller than the expected accuracy of the computation. Using this remark, we implement an adaptive spatial and temporal multilevel algorithm which treats differently the small- and large-scale components of the flow. We derive several a priori estimates in order to study the perturbation introduced into the approximated equations. All the interaction terms between small- and large-scale structures are frozen during several time steps. Finally, we implement the multilevel method in order to simulate a bidimensional flow described by the Burgers equations. We perform a parametric study of the procedure and its efficiency. The gain in CPU time is significant and the trajectories computed by our multi-scale method remain close to the trajectories obtained with the classical Galerkin method.
ISSN: 0168-9274

hal-00730710
https://hal.archives-ouvertes.fr/hal-00730710

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
In this work, we point out and apply some ideas for the numerical simulation by multi- level schemes of several wave equations including dispersive (Korteweg de Vries) and dissipative ones (Kuramoto-Sivashinski). We start from the approach developed for dissipative equations decomposing the solution into large (Y) and small (Z) scales, and we show that it is necessary to have a regularization in time to allow stable numerical treatment of Y and Z which are here generated using hierarchical processes such as Wavelets, Incremental Unknowns or Fourier.
ISSN: 1270-900X

hal-00768332
https://hal.archives-ouvertes.fr/hal-00768332
https://hal.archives-ouvertes.fr/hal-00768332/document
https://hal.archives-ouvertes.fr/hal-00768332/file/esaim_proc_2009_v27.pdf

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time discretisation of differential systems. In this article the authors propose generating numerical methods in numerical linear algebra by modelling the linear system to be solved as a given state of a dynamical system; the solution can be reached asymptotically, as a (asymptotically stable) steady state, but also as a finite time (shooting methods). In that way, any (stable) numerical scheme for the integration of such a problem can be presented as a method for solving linear systems. The authors discuss aspects of this approach, which allows them to recover some known methods but also to introduce new ones. Finally, some convergence results and numerical illustrations are presented.
ISSN: 1017-1398

hal-00730194
https://hal.archives-ouvertes.fr/hal-00730194

Éditeur(s) : HAL CCSD Wiley
Résumé : International audience
We study if the multilevel algorithm introduced by Debussche, T. Dubois and R. Temam [Theoret. Comput. Fluid Dynam. 7 (1995), no. 4, 279-315; Zbl 838.76060] and Dubois, F. Jauberteau and Temam [J. Sci. Comput. 8 (1993), no. 2, 167-194; MR1242960 (94f:65098)] for the 2D Navier-Stokes equations with periodic boundary conditions and spectral discretization can be generalized to more general boundary conditions and to finite elements. We first show that a direct generalization, as in [C. Calgaro, J. Laminie and R. Temam, Appl. Numer. Math. 23 (1997), no. 4, 403-442; MR1453424 (98d:76124)], for the Burgers equation, would not be very efficient. We then propose a new approach where the domain of integration is decomposed into subdomains. This enables us to define localized small-scale components and we show that, in this context, there is a good separation of scales. We conclude that all the ingredients necessary for the implementation of the multilevel algorithm are present.
ISSN: 0271-2091

hal-00730709
https://hal.archives-ouvertes.fr/hal-00730709

Éditeur(s) : HAL CCSD Wiley
Résumé : International audience
In this paper, the generalized Stokes problem αu−νΔu+∇p=f, ∇u=0, is considered, where ν denotes the kinematic viscosity, α is a positive parameter proportional to the inverse of the time-step, and the source term f gathers the external forcing, the nonlinear term, and the explicit part of the linear term. The system is derived from a time discretization for the Navier-Stokes equation governing the motion of a two-dimensional incompressible fluid. A domain decomposition numerical method is developed for solving the system. The original domain is divided into several nonoverlapping subdomains, and in each subdomain the concerned equations are discretized by certain finite element methods. A new feature of the method is that the pressure p is required to be piecewise linear rather than piecewise constant, so that the nonphysical oscillations in the solution using piecewise linear pressure are eliminated. However, a difficulty in using piecewise linear pressure is that the Stokes problem in each subdomain involves Dirichlet boundary conditions on pressure at the interface, which is not a real Stokes problem. To avoid the difficulty, a modification is proposed which removes the continuity conditions on the pressure across the interface. In this way, the Stokes problem in each subdomain has only Dirichlet boundary conditions on the velocity. Formulations and discussions for both the methods are presented, which shows that the modified method is simpler than the original one. Numerical experiments are displayed to show the efficiency of the methods, especially that the modified method has the same accuracy as the original one.
ISSN: 0749-159X

hal-00730200
https://hal.archives-ouvertes.fr/hal-00730200

Éditeur(s) : HAL CCSD
Résumé : Nous proposons dans cet article de mettre en avant quelques idées pour la simulation numérique de plusieurs classes d'équations d'ondes dispersives (KdV, Benjamin-Ono) ou non (Kuramoto-Sivashinski) par des méthodes multiniveaux. Nous partons des idées initialement développés pour les équations dissipatives et en proposons une extension. Il en ressort que la présence d'une régularisation en temps est nécessaire pour pouvoir traiter de manière stable, et avec des schémas appropriés, les différentes échelles de la solution, préalablement générées à l'aide de méthodes hierarchiques (inconnues incrémentales, ondelettes, Fourier)
https://hal.inria.fr/inria-00271154

inria-00271154
https://hal.inria.fr/inria-00271154
https://hal.inria.fr/inria-00271154/document
https://hal.inria.fr/inria-00271154/file/cclz.pdf