High-order dimensionally split Lagrange-remap schemes for ideal magnetohydrodynamics
Marc Wolff Stéphane Jaouen Hervé Jourdren Eric Sonnendrücker
Discrete & Continuous Dynamical Systems - S 2012, 5(2): 345-367 doi: 10.3934/dcdss.2012.5.345
We first propose a new class of high-order finite volume schemes for solving the 1-D ideal magnetohydrodynamics equations that is particularly well-suited for modern computer architectures. Applicable to arbitrary equations of state, these schemes, based on a Lagrange-remap approach, are high-order accurate in both space and time in the non-linear regime. A multidimensional extension on 2-D Cartesian grids using a high-order dimensional splitting technique is then proposed. Numerical results up to fourth-order on smooth and non-smooth test problems are also provided.
keywords: dimensional splitting. High-order schemes ideal magnetohydrodynamics
Hervé Guillard Eric Sonnendrücker
Discrete & Continuous Dynamical Systems - S 2012, 5(2): i-i doi: 10.3934/dcdss.2012.5.2i
This special issue of DCDS-S is dedicated to the proceedings of the second edition of the Conference Numerical Models for Controlled Fusion (NMCF09) that was held on the Island of Porquerolles (France) April 20-24, 2009. “

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Guiding-center simulations on curvilinear meshes
Jean-Philippe Braeunig Nicolas Crouseilles Michel Mehrenberger Eric Sonnendrücker
Discrete & Continuous Dynamical Systems - S 2012, 5(2): 271-282 doi: 10.3934/dcdss.2012.5.271
The purpose of this work is to design simulation tools for magnetised plasmas in the ITER project framework. The specific issue we consider is the simulation of turbulent transport in the core of a Tokamak plasma, for which a 5D gyrokinetic model is generally used, where the fast gyromotion of the particles in the strong magnetic field is averaged in order to remove the associated fast time-scale and to reduce the dimension of 6D phase space involved in the full Vlasov model. Very accurate schemes and efficient parallel algorithms are required to cope with these still very costly simulations. The presence of a strong magnetic field constrains the time scales of the particle motion along and accross the magnetic field line, the latter being at least an order of magnitude slower. This also has an impact on the spatial variations of the observables. Therefore, the efficiency of the algorithm can be improved considerably by aligning the mesh with the magnetic field lines. For this reason, we study the behavior of semi-Lagrangian solvers in curvilinear coordinates. Before tackling the full gyrokinetic model in a future work, we consider here the reduced 2D Guiding-Center model. We introduce our numerical algorithm and provide some numerical results showing its good properties.
keywords: plasma curvilinear coordinates guiding-center. Semi-Lagrangian scheme simulation
An exponential integrator for a highly oscillatory vlasov equation
Emmanuel Frénod Sever A. Hirstoaga Eric Sonnendrücker
Discrete & Continuous Dynamical Systems - S 2015, 8(1): 169-183 doi: 10.3934/dcdss.2015.8.169
In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution.
keywords: highly oscillatory ODEs exponential time differencing. Particle-in-Cell method Vlasov-Poisson system

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