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

Erwan Deriaz; Bruno Després; Gloria Faccanoni; Kirill Gostaf; Lise-Marie Imbert-Gérard; Georges Sadaka; Remy Sart

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.

Document type :

Journal articles

Domain :

Mathematics [math] / Numerical Analysis [math.NA]

Mathematics [math] / Analysis of PDEs [math.AP]

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