Magnetic equations with FreeFem++: The Grad-Shafranov equation & the current hole

¹ Laboratoire de Mécanique, Modélisation et Procédés Propres, 38 rue Frédéric Joliot-Curie 13451 Marseille, France
e-mail: erwan.deriaz@l3m.univ-mrs.fr
² Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris, France
e-mail: despres@ann.jussieu.fr & gostaf@ann.jussieu.fr & imbert@ann.jussieu.fr
³ Institut de mathématiques de Toulon et du Var, Université du Sud Toulon-Var, 83957 La Garde, France
e-mail: faccanon@univ-tln.fr
⁴ Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens, France
e-mail: georges.sadaka@u-picardie.fr
⁵ École Supérieure d’Ingénieurs Léonard de Vinci, 92916 Paris, France
e-mail: remy.sart@devinci.fr

Abstract

FreeFem++ [11] 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.