# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

April 2012 , Volume 5 , Issue 2

Issue dedicated to the proceedings of the second edition of
the Conference Numerical Models for Controlled Fusion (NMCF09)

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2012, 5(2): i-i doi: 10.3934/dcdss.2012.5.2i +[Abstract](705) +[PDF](74.9KB)
Abstract:
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. “

2012, 5(2): 245-255 doi: 10.3934/dcdss.2012.5.245 +[Abstract](882) +[PDF](390.3KB)
Abstract:
The present work concerns the numerical simulations of radiative transfer. To address such an issue, the $M1$-model is here adopted. Indeed, this moment model is known to preserve several essential physical properties about radiative energy and radiative flux. In addition, it reduces drastically the numerical cost of the simulations. Unfortunately, the model is not able to restore the expected diffusive regime as prescribed by physics. To correct such a failure, a suitable numerical procedure is derived. The proposed approximation technique enforces, in a sense to be specified, a numerical diffusive regime governed by the Rosseland's mean value of the opacity as imposed by the radiative transfer equation. Numerical experiments issuing from relevant physical benchmarks, illustrate the interest of the derived method.
2012, 5(2): 257-269 doi: 10.3934/dcdss.2012.5.257 +[Abstract](807) +[PDF](357.0KB)
Abstract:
One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion requires the confinement of the plasma into a bounded domain and for this, we appeal to the magnetic confinement. Several models exist for describing the evolution of strongly magnetized plasmas. The subject matter of this paper is to provide a rigorous derivation of the guiding-center approximation in the general three dimensional setting, under the action of large stationary inhomogeneous magnetic fields.
2012, 5(2): 271-282 doi: 10.3934/dcdss.2012.5.271 +[Abstract](963) +[PDF](1192.3KB)
Abstract:
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.
2012, 5(2): 283-305 doi: 10.3934/dcdss.2012.5.283 +[Abstract](1114) +[PDF](515.6KB)
Abstract:
We propose a second order finite volume scheme to discretize the one-dimensional Vlasov-Poisson system with boundary conditions. For this problem, a rather general initial and boundary data lead to a unique solution with bounded variations but such a solution becomes discontinuous when the external voltage is large enough. We prove that the numerical approximation converges to the weak solution and show the efficiency of the scheme to simulate beam propagation with several particle species.
2012, 5(2): 307-328 doi: 10.3934/dcdss.2012.5.307 +[Abstract](976) +[PDF](907.6KB)
Abstract:
The problems associated to the optimization of the systems of a future fusion reactor require new developments due to the limits of existing models, which are unable to describe the experimental behaviour of diagnostics with the required accuracy. This is also true for the study of the coupling antenna-plasma or the computations of the deposits of power for the plasma heating. Simulations on wave propagation on full ITER size is a key issue properly to take into account all the possible effects arising during the wave propagation. These effects should be the scattering processes, back- and forward- scattering, absorption, multi-reflections, diffraction, interference, depolarisation, and mode conversion. Each phenomenon requires an adapted description having its own numerical conditions, which are functions of mesh size, density of modes associated to given plasma fluctuations to name a few. The numerical requirements to fulfil the theoretical modelling cannot always be reached especially when all the space dimensions are needed to have a realistic description of the wave propagation in fluctuating plasmas. A rapid review of each model with its limitations and the specific tools associated to the different kinds of reflectometry diagnostics is detailed. A discussion on the problems and the works underway in the plasma reflectometry community concludes this review.
2012, 5(2): 329-343 doi: 10.3934/dcdss.2012.5.329 +[Abstract](924) +[PDF](893.5KB)
Abstract:
We are concerned here with the modelling of the laser propagation and its interaction with a plasma. In a first part, we recall first some features related to the paraxial approximation of the solution of the wave equation and the coupling model between the plasma hydrodynamics and the laser propagation. In a second part, we consider the coupling with the ion acoustic waves which has to be accounted to model the Brillouin instability. It leads to the three-wave coupling problem which is a crucial in plasma physics. We give some mathematical properties of this system specially in the case when the speed of light is assumed to be infinite. We also propose a numerical method based on a implicit time discretization. It is illustrated on test cases.
2012, 5(2): 345-367 doi: 10.3934/dcdss.2012.5.345 +[Abstract](1341) +[PDF](1442.5KB)
Abstract:
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.

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