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-DES (ex-DEN) - CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire)
4 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
5 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
6 ESILV - Ecole Supérieure d'Ingénieurs Léonard de Vinci

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.

Domains: Mathematics [math]; Numerical Analysis [math.NA]; Mathematics [math]; Analysis of PDEs [math.AP]

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, 2011, 32, pp.76-94. ⟨10.1051/proc/2011013⟩. ⟨hal-00976962⟩