Issue ESAIM: Proc. Volume 16, 2007 CEMRACS 2005 - Computational Aeroacoustics and Computational Fluid Dynamics in Turbulent Flows Page(s) 195 - 210 DOI http://dx.doi.org/10.1051/proc:2007013 Published online 02 March 2007

ESAIM: Proc., 2007, Vol. 16, pp. 195-210
DOI: 10.1051/proc:2007013

Wavelet denoising for postprocessing of a 2D Particle - In - Cell code

Salimou Gassama¹, Éric Sonnendrücker¹, Kai Schneider², Marie Farge³ and Margarete O. Domingues<sup>4

1</sup>  Institut de Recherche Mathématique Avancée, UMR CNRS 7501 Université Louis Pasteur, Strasbourg, France.
²  MSNM-CNRS & CMI, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France.
³  LMD-CNRS, Ecole Normale Supérieure, 24 rue Lhomond,75231 Paris cedex 05, France.
⁴  Laboratório Associado de Computação e Matemática Aplicada (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), Av. dos Astronautas, 1758, 12227-010 São José dos Campos, Brasil.

(Published online: 2 March 2007)

Abstract
In this paper, we aim at improving the accuracy of a Vlasov-Poisson solver using a 2D Particle-In-Cell (PIC) scheme, by denoising its charge density field. To this end, we have used an improvement of Donoho and Johnstone's wavelet denoising technique. To some extent, our work is a continuation of that performed by Chehab et al. [Chehab, Cohen, Roche, Jennequin, Nieto and Roland, CEMRACS 2003, IRMA Lectures in Mathematics and Theoretical Physics, EMS, pp. 29-42]. Indeed, they made such a study in the one dimensional case and validated their analysis by considering the simulation of the Landau damping phenomenon. They concluded on the efficiency of the method in reducing the number of particles. However, our approach is quite different, since we do not use wavelets to directly interpolate the charge density, but we smooth the density field calculated by the PIC code. This is carried out via an iterative wavelet denoising technique introduced by Azzalini et al. [Azzalini, Farge and Schneider, Appl. Comput. Harm. Anal. 18 (2005), no. 2, 177185]. Our work consists in studying the application of the method as a post-processing tool, in view of a future embedding into the PIC code. The results are the following: first, we showed that the hypotheses underlying the application of this method are valid. Secondly, we can infer from this study that it is possible to significantly reduce the amount of data needed for a simulation.

Résumé
L'objectif de ces travaux est d'accroître la précision d'un solveur Vlasov-Poisson basé sur un schéma de type Particle-In-Cell (PIC). Pour ce faire, à l'instar de Chehab et coll. [Chehab, Cohen, Roche, Jennequin, Nieto and Roland, CEMRACS 2003, IRMA Lectures in Mathematics and Theoretical Physics, EMS, pp. 29-42], nous avons choisi d'utiliser des techniques d'analyse multirésolution par ondelettes pour débruiter la densité de charge. Cependant, à la différence de ces auteurs, nous effectuons un post-traitement de la densité de charge en appliquant une technique itérative mise au point par Azzalini et coll. [Azzalini, Farge and Schneider, Appl. Comput. Harm. Anal. 18 (2005), no. 2, 177185]. Cette étude est un préalable à l'implémentation d'un algorithme de débruitage dynamique du code PIC. Les expériences numériques menée sur le cas-test de l'amortissement Landau nous ont permi de valider les hypothèses sous-jacentes et portent à croire qu'il est possible de réduire de manière significative le nombre de particules nécessaires pour une précision donnée.

Mathematics Subject Classification. 65M

Key words: Wavelet, denoising, Particle-in-cell, plasma, Vlasov equation


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