Finite Volume Approximation of MHD Equations with Euler Potential

Abstract : The possibility to produce energy by fusion reactions is being studied in experimental devices called tokamaks where charged particles are confined in a toroidal vacuum chamber thanks to a very large magnetic field. The ITER device currently build in Cadarache (France) will be the largest tokamak ever realized for these experiments. The large scale dynamics of charged particles in a tokamak as ITER can be described by the Magneto-HydroDynamics equations (MHD). This system of equations contains as an involution the divergence-free constraint of the magnetic field, div B = 0 that has to be maintained by the numerical approximation. The respect of the divergence-free constraint can be achieved in two different ways. The first class consists in adding to the MHD system penalization terms ensuring that the magnetic field will be solenoidal. The second class consists in formulating the MHD system in term of the vectorial potentiel A that satisfies curl A = B, fulfilling thus automatically the divergence-free constraint. The proposed method is a formulation of the MHD system in term of the mixture of the two former classes. The resulting system is divergence-free constraint preserving and can be approximated by standard Finite Volume methods. Various numerical tests on well-known standard problems in MHD show that this approach is an interesting alternative and opens possibility to use a conservative formulation based on B of the MHD system.