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Unconditionally stable numerical simulations of a new generalized reduced resistive magnetohydrodynamics model. (English) Zbl 1455.76209

Summary: Reduced-resistive magnetohydrodynamics (MHD) models are used in understanding different phenomenon in various domains, for example, astrophysics to model magnetotail or for solar arcades [J. M. Finn and P. N. Guzdar, “Loss of equilibrium and reconnection in tearing of two dimensional equilibrias”, AIP Phys. Fluids B 5, 2870–2876 (1993; doi:10.1063/1.860674)], modeling plasma confinements in reverse field pinch [H. R. Strauss, Phys. Fluids 28, 2786–2792 (1985; Zbl 0572.76117)] and tokamaks [H. R. Strauss, “Reduced MHD in nearly potential magnetic fields”, J. Plasma Phys. 57, No. 1, 83–87 (1997; doi:10.1017/S0022377896005296); J. P. Freidberg, Plasma physics and fusion energy. Cambridge: Cambridge University Press (2008; doi:10.1017/CBO9780511755705)]. In this context, recently, a new generalized reduced-resistive MHD model, which can make use of an arbitrary density profile was proposed [B. Després and R. Sart, ESAIM, Math. Model. Numer. Anal. 46, No. 5, 1081–1106 (2012; Zbl 1267.76034)]. We in this work show that this proposed theoretical model can be realized numerically as well, and that it is very robust if the equation set is written in a very particular form using the properties of FEM. To illustrate these points, we pick the current hole configuration [T. Fujita et al., “Plasma equilibrium and confinement in a tokamak with nearly zero central current density in JT-60U”, Phys. Rev. Lett. 87, No. 11, Article ID 245001, 4 p. (2001; doi:10.1103/PhysRevLett.87.245001); N. C. Hawkes et al., “Observation of zero current density in the core of JET discharges with lower hybrid heating and current drive”, ibid. 87, No. 11, Article ID 115001, 4 p. (2001; doi:10.1103/PhysRevLett.87.115001)], which was modeled using reduced-resistive MHD and remodel it using different combinations of current sources and density profiles. Our model can be implemented with reasonable computational resources at the price of solving a well-posed global linear system and it is unconditionally stable. These features are also demonstrated as a part of our numerical experiments.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
82D10 Statistical mechanics of plasmas
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