Magnetic Equations with FreeFem++: the Grad-Shafranov Equation & the Current Hole

Erwan Deriaz 1 Bruno Després 2 Gloria Faccanoni 3, 4 Kirill Gostaf 2 Lise-Marie Imbert-Gérard 2 Georges Sadaka 5 Remy Sart 6
1 LMD - Laboratoire de Météorologie Dynamique (UMR 8539)
2 LJLL - Laboratoire Jacques-Louis Lions
3 CEA-DEN - CEA-Direction de l'Energie Nucléaire
4 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
5 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
6 ESI Léonard de Vinci - Ecole Supérieure d'Ingénieurs Léonard de Vinci
Abstract : FreeFem++ is a software for the numerical solution of partial differential equations. It is based on finite element method. The FreeFem++ platform aims at facilitating teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. Our goal in this work is to evaluate the FreeFem++ tool on basic magnetic equations arising in Fusion Plasma in the context of the ITER project. First we consider the Grad-Shafranov equation, which is derived from the static ideal MHD equations assuming axisymetry. Some of the properties of the equation and its analytical solutions are discussed. Second we discretize a reduced resistive MHD model which admits solutions of the Grad-Shafranov equation as stationary solutions. Then the physical stability of these stationary solutions is investigated through numerical experiments and the numerical stability of the algorithm is discussed.
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ESAIM: Proceedings, EDP Sciences, 2011, 32, pp.76-94. 〈10.1051/proc/2011013〉
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Erwan Deriaz, Bruno Després, Gloria Faccanoni, Kirill Gostaf, Lise-Marie Imbert-Gérard, et al.. Magnetic Equations with FreeFem++: the Grad-Shafranov Equation & the Current Hole. ESAIM: Proceedings, EDP Sciences, 2011, 32, pp.76-94. 〈10.1051/proc/2011013〉. 〈hal-00976962〉

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