Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code
Franck, Emmanuel1 ; Hölzl, Matthias2 ; Lessig, Alexander2 ; Sonnendrücker, Eric2, 3
1 Inria Nancy grand Est, TONUS Team, 67000 Strasbourg, France.
2 Max-Planck-Institut für Plasmaphysik, Boltzmannstr. 2, 85748 Garching, Germany.
3 Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1331-1365.

In this paper we present a rigorous derivation of the reduced MHD models with and without parallel velocity that are implemented in the non-linear MHD code JOREK. The model we obtain contains some terms that have been neglected in the implementation but might be relevant in the non-linear phase. These are necessary to guarantee exact conservation with respect to the full MHD energy. For the second part of this work, we have replaced the linearized time stepping of JOREK by a non-linear solver based on the Inexact Newton method including adaptive time stepping. We demonstrate that this approach is more robust especially with respect to numerical errors in the saturation phase of an instability and allows to use larger time steps in the non-linear phase.

Reçu le :
DOI : 10.1051/m2an/2015014
Classification : 65H10, 35Q35, 35Q60, 76W05
Mots-clés : MHD, instabilities, nonlinear solvers, reduction, toroidal
Franck, Emmanuel 1 ; Hölzl, Matthias 2 ; Lessig, Alexander 2 ; Sonnendrücker, Eric 2, 3

1 Inria Nancy grand Est, TONUS Team, 67000 Strasbourg, France.
2 Max-Planck-Institut für Plasmaphysik, Boltzmannstr. 2, 85748 Garching, Germany.
3 Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany. 
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     title = {Energy conservation and numerical stability for the reduced {MHD} models of the non-linear {JOREK} code},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
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Franck, Emmanuel; Hölzl, Matthias; Lessig, Alexander; Sonnendrücker, Eric. Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1331-1365. doi : 10.1051/m2an/2015014. http://archive.numdam.org/articles/10.1051/m2an/2015014/

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