CIRM - Videos & books Library - Highly-oscillatory evolution equations: averaging and numerics

Virtualconference

Auteurs : Lemou, Mohammed (Auteur de la Conférence)
CIRM (Editeur )

Loading the player...

Résumé : Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to capture low frequencies and transition regimes. We present (very briefly) two strategies allowing to remove this obstacle for a large class of evolution problems : a 2-scale method and a micro/macro method. Two different frameworks will be considered : constant frequency, and variable - possibly vanishing - frequency. The result of these approaches is the construction of numerical schemes whose order of accuracy no longer depends on the frequency of oscillation, one then speaks of uniform accuracy (UA) for these schemes. Finally, a new technique for systematizing these two methods will be presented. Its purpose is to reduce the number of inputs that the user must provide to apply the method in practice. In other words, only the values of the field defining the evolution equation (and not its derivatives) are used.These methods have been successfully applied to solve a number of evolution models: non-linear Schrödinger and Klein-Gordon equations, Vlasov-Poisson kinetic equation with strong magnetic field, quantum transport in graphene.

Keywords : multi-scale numerical methods; highly oscillatory equations; averaging; varying frequency

Codes MSC :
35Q55 - NLS-like equations (nonlinear Schrödinger)
37L05 - General theory, nonlinear semigroups, evolution equations
65L05 - Initial value problems for ODE (numerical method)

Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Mohammed_LEMOU.pdf

Informations sur la Vidéo

Réalisateur :

Hennenfent, Guillaume

Langue :

Date de publication :

Date de captation :

Sous collection :

arXiv category :

Domaine :

Format :

Durée :

Audience :

Download :

https://videos.cirm-math.fr/2021-03-22_Lemou.mp4

Informations sur la Rencontre

Nom de la rencontre : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et Analyse
Organisateurs de la rencontre : Bostan, Mihaï ; Jin, Shi ; Mehrenberger, Michel ; Montibeller, Celine
Dates : 22/03/2021 - 26/03/2021
Année de la rencontre : 2021
URL Congrès : https://www.chairejeanmorlet.com/2355.html

Données de citation

DOI : 10.24350/CIRM.V.19735003
Citer cette vidéo: Lemou, Mohammed (2021). Highly-oscillatory evolution equations: averaging and numerics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19735003
URI : http://dx.doi.org/10.24350/CIRM.V.19735003

Voir aussi

[Virtualconference] Model predictive and random batch methods for a guiding problem / Auteur de la Conférence Zuazua, Enrique.
[Virtualconference] A De Giorgi argument for $L^{\infty}$ solution to the Boltzmann equation without angular cutoff / Auteur de la Conférence Yang, Tong.
[Virtualconference] An asymptotic preserving method for Levy Fokker Planck equation with fractional diffusion limit / Auteur de la Conférence Wang, Li.
[Virtualconference] Bacterial movement by run and tumble: models, patterns, pathways, scales / Auteur de la Conférence Perthame, Benoît.
[Virtualconference] Quantitative De Giorgi methods in kinetic theory / Auteur de la Conférence Mouhot, Clément.
[Virtualconference] Stable and unstable steady states for the HMF model / Auteur de la Conférence Mehats, Florian.
[Virtualconference] An inverse problem for a model of cell motion and chemotaxis / Auteur de la Conférence Klingenberg, Christian.
[Virtualconference] Mean-field-type limits of interacting particle systems for multiple species / Auteur de la Conférence Juengel, Ansgar.
[Virtualconference] Large stochastic systems of interacting particles / Auteur de la Conférence Jabin, Pierre-Emmanuel.
[Virtualconference] A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation / Auteur de la Conférence Hu, Jingwei.
[Virtualconference] Stabilization of random kinetic equations / Auteur de la Conférence Herty, Michael.
[Virtualconference] Classical and quantum particles coupled to a vibrational environment / Auteur de la Conférence Goudon, Thierry.
[Virtualconference] Partial regularity in time for the Landau equation with Coulomb interaction / Auteur de la Conférence Golse, François.
[Virtualconference] Dynamical low-rank approximation for radiation transport / Auteur de la Conférence Frank, Martin.
[Virtualconference] Large time asymptotics for evolution equations with mean field couplings / Auteur de la Conférence Dolbeault, Jean.
[Virtualconference] Quasilinear approximation of Vlasov and Liouville equations / Auteur de la Conférence Bardos, Claude.
[Virtualconference] Collective dynamics of quantized vortices in superfluidity and superconductivity / Auteur de la Conférence Bao, Weizhu.
[Virtualconference] Modal based hypocoercivity methods on the torus and the real line with application to Goldstein-Taylor models / Auteur de la Conférence Arnold, Anton.

Bibliographie

CHARTIER, Philippe, LEMOU, Mohammed, MÉHATS, Florian, et al. A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations. Foundations of Computational Mathematics, 2020, vol. 20, no 1, p. 1-33. - https://doi.org/10.1007/s10208-019-09413-3

CHARTIER, Ph, LEMOU, Mohammed, MÉHATS, Florian, et al. Highly oscillatory problems with time-dependent vanishing frequency. SIAM Journal on Numerical Analysis, 2019, vol. 57, no 2, p. 925-944. - https://doi.org/10.1137/18M1203456

CROUSEILLES, Nicolas, JIN, Shi, LEMOU, Mohammed, et al. A micro-macro method for a kinetic graphene model in one space dimension. Multiscale Modeling & Simulation, 2020, vol. 18, no 1, p. 444-474. - https://doi.org/10.1137/18M1173770

CHARTIER, Philippe, LEMOU, Mohammed, MÉHATS, Florian, et al. Derivative-free high-order uniformly accurate schemes for highly-oscillatory systems. To appear 2021. - https://hal.inria.fr/hal-03141156

Lemou.jpg

Sélection Signaler une erreur

TUTELLES

PARTENAIRES

Destination de la recherche

Raccourcis

Voir aussi