[{"command":"insert","method":null,"selector":".course-modal-ajax-placeholder-958183","data":"\u003Cdiv class=\u0022container content-rte\u0022\u003E\n  \u003Cdiv class=\u0022row\u0022\u003E\n    \u003Cdiv class=\u0022box col-md-6 bg-primary text-white\u0022\u003E\n      \u003Ch2 class=\u0022h1 text-left\u0022\u003E  M\u00e9thodes Num\u00e9riques pour les \u00e9quations aux d\u00e9riv\u00e9es partielles - Site ENS\n\u003C\/h2\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EECTS :\n          3\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ESemester:\n          Semestre 1\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ED\u00e9tail du volume horaire\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ELecture:\u003C\/span\u003E\n                                      \u003Cspan\u003E14\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003EPractical study :\u003C\/span\u003E\n                                      \u003Cspan\u003E8\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003EDirected study:\u003C\/span\u003E\n                                      \u003Cspan\u003E8\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                                            \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ELangue d\u0026#039;enseignement\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EFran\u00e7ais\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EEnseignement \u00e0 distance\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003Enon\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n    \u003Cdiv class=\u0022box col-md-6\u0022\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EPr\u00e9requis\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003E\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Cours d\u0027analyse num\u00e9rique de licence de m\u00e9canique ou UE \u00e9quivalentes (introduction sur les \u00e9quations aux d\u00e9riv\u00e9es partielles, formulation variationnelle, th\u00e9or\u00e8me de Lax-Milgram\u2026).\u003Cbr\u003E\n\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Cours de m\u00e9thodes num\u00e9riques de licence de m\u00e9canique ou UE \u00e9quivalentes (r\u00e9solution des syst\u00e8mes lin\u00e9aires, m\u00e9thodes directes, m\u00e9thodes it\u00e9ratives\u2026).\u003Cbr\u003E\n\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Notions de programmation.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EProgramme \/ plan \/ contenus\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EChapitre 1 : Rappels sur les \u00e9quations aux d\u00e9riv\u00e9es partielles, notion de recherche de solution num\u00e9rique\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;D\u00e9finition et origine des EDP en physique et m\u00e9canique des milieux continus.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Classification des EDP : elliptiques, paraboliques, hyperboliques.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Conditions aux limites et conditions initiales : r\u00f4le et typologie (Dirichlet, Neumann, mixte).\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Notion de solution exacte vs solution num\u00e9rique.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Introduction \u00e0 la discr\u00e9tisation et \u00e0 la n\u00e9cessit\u00e9 des m\u00e9thodes num\u00e9riques pour les probl\u00e8mes complexes.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Pr\u00e9sentation des crit\u00e8res d\u2019\u00e9valuation des sch\u00e9mas num\u00e9riques : consistance, stabilit\u00e9, convergence.\u003C\/p\u003E\n\u003Cp\u003EChapitre 2 : M\u00e9thode des diff\u00e9rences finies\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Principe g\u00e9n\u00e9ral de la m\u00e9thode : approximation des d\u00e9riv\u00e9es par des diff\u00e9rences sur un maillage r\u00e9gulier.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Sch\u00e9mas explicites et implicites pour les EDP temporelles.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Analyse de la consistance et de la stabilit\u00e9 des sch\u00e9mas (ex. crit\u00e8re de Courant-Friedrichs-Lewy pour les \u00e9quations hyperboliques).\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Applications \u00e0 des probl\u00e8mes mod\u00e8les : \u00e9quation de la chaleur, \u00e9quation d\u2019onde, \u00e9quation de Laplace.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Limites et extensions : sch\u00e9mas d\u2019ordre sup\u00e9rieur, maillages non uniformes.\u003C\/p\u003E\n\u003Cp\u003EChapitre 3 : M\u00e9thode des \u00e9l\u00e9ments finis\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Principe de la m\u00e9thode : formulation variationnelle et approximation des solutions sur des \u00e9l\u00e9ments finis.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Construction du maillage et choix des fonctions de forme.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Assemblage des matrices globales et r\u00e9solution des syst\u00e8mes lin\u00e9aires associ\u00e9s.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Analyse de la convergence et de l\u2019erreur des solutions approximatives.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Applications \u00e0 des probl\u00e8mes elliptiques et paraboliques simples.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Avantages de la m\u00e9thode pour les g\u00e9om\u00e9tries complexes et les conditions aux limites vari\u00e9es.\u003C\/p\u003E\n\u003Cp\u003EChapitre 4 : M\u00e9thodes des volumes finis\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Principe g\u00e9n\u00e9ral : conservation locale des flux sur des volumes de contr\u00f4le.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Discr\u00e9tisation des \u00e9quations int\u00e9grales et passage \u00e0 des formulations discr\u00e8tes.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Traitement des termes de convection et diffusion.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Analyse de la stabilit\u00e9 et de la pr\u00e9cision des sch\u00e9mas.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Applications \u00e0 des probl\u00e8mes de transport, m\u00e9canique des fluides et transfert de chaleur.\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Comparaison avec les m\u00e9thodes des diff\u00e9rences finies et des \u00e9l\u00e9ments finis : avantages et limites.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EObjectifs d\u0026#039;apprentissage\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003ECe cours apportera aux \u00e9tudiantes et aux \u00e9tudiants les connaissances de base n\u00e9cessaires pour comprendre et classifier les \u00e9quations aux d\u00e9riv\u00e9es partielles issues de la m\u00e9canique des milieux continus, ainsi que pour relier ces mod\u00e8les physiques \u00e0 leur formulation math\u00e9matique. Il leur permettra de ma\u00eetriser les principes fondamentaux des m\u00e9thodes num\u00e9riques de r\u00e9solution, notamment les diff\u00e9rences finies, les volumes finis et les \u00e9l\u00e9ments finis, et d\u2019analyser la consistance, la stabilit\u00e9 et la convergence des sch\u00e9mas associ\u00e9s. Les \u00e9tudiantes et \u00e9tudiants apprendront \u00e0 mettre en \u0153uvre ces m\u00e9thodes \u00e0 travers des simulations num\u00e9riques de probl\u00e8mes mod\u00e8les et \u00e0 d\u00e9velopper un regard critique sur le choix et l\u2019application des m\u00e9thodes num\u00e9riques selon la nature du probl\u00e8me et les contraintes de calcul.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EOrganisation g\u00e9n\u00e9rale et modalit\u00e9s p\u00e9dagogiques\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003ELe cours abordera les points suivants :\u003Cbr\u003E\n\u2022 Pr\u00e9sentation, classification et analyse d\u0027\u00e9quations aux d\u00e9riv\u00e9es partielles mod\u00e8les, issues de la physique et\/ou de la m\u00e9canique des milieux continus.\u003Cbr\u003E\n\u2022 Introduction des diff\u00e9rentes m\u00e9thodes num\u00e9riques de r\u00e9solution des \u00e9quations aux d\u00e9riv\u00e9es partielles et \u00e9l\u00e9ments d\u2019analyse pour caract\u00e9riser ces m\u00e9thodes (consistance, stabilit\u00e9, convergence\u2026) : m\u00e9thode des diff\u00e9rences finies, m\u00e9thode des volumes finis, m\u00e9thode des \u00e9l\u00e9ments finis.\u003C\/p\u003E\n\u003Cp\u003ELe cours sera accompagn\u00e9 de travaux dirig\u00e9s et de s\u00e9ances de TP. Ces derni\u00e8res seront l\u2019occasion de coder les m\u00e9thodes num\u00e9riques introduites pour les simulations num\u00e9riques de quelques probl\u00e8mes mod\u00e8les.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EModalit\u00e9s p\u00e9dagogiques particuli\u00e8res\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003ELe cours combine cours magistral et travaux dirig\u00e9s pour permettre l\u2019acquisition des concepts th\u00e9oriques et leur mise en pratique. Des s\u00e9ances de travaux pratiques (TP) sont pr\u00e9vues afin de coder et de simuler les m\u00e9thodes num\u00e9riques \u00e9tudi\u00e9es sur des probl\u00e8mes mod\u00e8les. Les \u00e9tudiants sont encourag\u00e9s \u00e0 travailler en petits groupes lors des TP pour favoriser la collaboration et l\u2019\u00e9change d\u2019id\u00e9es.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EComp\u00e9tences\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EVoici une liste de comp\u00e9tences que les \u00e9tudiants d\u00e9velopperont dans ce cours :\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Identifier et classer les principales \u00e9quations aux d\u00e9riv\u00e9es partielles (elliptiques, paraboliques, hyperboliques) issues de la m\u00e9canique des milieux continus ;\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Comprendre les principes fondamentaux des m\u00e9thodes de discr\u00e9tisation des EDP (diff\u00e9rences finies, volumes finis, \u00e9l\u00e9ments finis) ;\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Mettre en \u0153uvre num\u00e9riquement ces m\u00e9thodes pour r\u00e9soudre des probl\u00e8mes mod\u00e8les ;\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Analyser la qualit\u00e9 d\u2019un sch\u00e9ma num\u00e9rique \u00e0 travers les notions de consistance, stabilit\u00e9 et convergence ;\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u00c9valuer le conditionnement et la pr\u00e9cision des syst\u00e8mes lin\u00e9aires issus de la discr\u00e9tisation ;\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Programmer et simuler des solutions num\u00e9riques \u00e0 l\u2019aide d\u2019outils de calcul scientifique ;\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Interpr\u00e9ter et valider les r\u00e9sultats num\u00e9riques en les confrontant aux solutions analytiques ou physiques attendues ;\u003Cbr\u003E\n\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;D\u00e9velopper une approche critique vis-\u00e0-vis du choix des m\u00e9thodes num\u00e9riques selon la nature du probl\u00e8me \u00e9tudi\u00e9.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EBibliographie\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003E\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;R. Th\u00e9odor, Initiation \u00e0 l\u0027analyse num\u00e9rique, CNAM cours A, Masson, 1994.\u003Cbr\u003E\n\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;A. Quarteroni, Numerical Models For Differential Problems, Springer Verlag, 2012.\u003Cbr\u003E\n\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;G. Allaire, Analyse Num\u00e9rique et Optimisation, Editions de l\u2019Ecole Polyechnique, 2012.\u003Cbr\u003E\n\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Stig Larsson \u0026amp; Vidar Thom\u00e9e. Partial Differential Equations with Numerical Methods. Springer 2003\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ENature de l\u0026#039;\u00e9valuation\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EEvaluation Continue non Int\u00e9grale\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n  \u003C\/div\u003E\n\u003C\/div\u003E\n","settings":null}]