[en] Highlights: • Kolmogorov linear stability problem solved by asymptotic matching. • Nonlinear steady state exhibits seemingly universal solutions. • Asymptotic-Preserving methods allow efficient integration to long times. • Asymptotic-Preserving schemes keep numerical accuracy at all times. We study the Kolmogorov model of a shear flow by means of a newly developed Asymptotic-Preserving method for the numerical resolution of the two-dimensional vorticity-Poisson (Navier-Stokes) system. The scheme is validated by comparing the results with those obtained with an explicit spectral code and with an analytic result about the linear instability regime. We show that the AP-properties of the method allow one to deal efficiently with the multi-scale nature of the problem by tuning the time step to the physical one and not by stability constraints. As a result, we investigate the long time scale evolution of the Kolmogorov flow, observing that it evolves into a final stable stationary state characterised by a seemingly universal relation between stream-function and vorticity.

[en] This paper deals with the numerical study of a strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to high anisotropy. Furthermore, the recently proposed Asymptotic-Preserving method (Lozinski et al., 2012) allows one to perform simulations regardless of the anisotropy strength but its application is limited to the case where the anisotropy direction is given by a field whose lines are all open. In this paper we introduce a new Asymptotic-Preserving method, which overcomes those limitations without any loss of precision or increase in computational costs. The convergence of the method is shown to be independent of the anisotropy parameter 0 ≤ ε ≤ 1 for fixed coarse Cartesian grids, and for variable anisotropy directions. The context of this work is magnetically confined fusion plasmas. (authors)

[en] This work is devoted to the study of field-aligned interpolation in semi-Lagrangian codes. In the context of numerical simulations of magnetic fusion devices, this approach is motivated by the observation that gradients of the solution along the magnetic field lines are typically much smaller than along a perpendicular direction. In toroidal geometry, field-aligned interpolation consists of a 1D interpolation along the field line, combined with 2D interpolations on the poloidal planes (at the intersections with the field line). A theoretical justification of the method is provided in the simplified context of constant advection on a 2D periodic domain: unconditional stability is proven, and error estimates are given which highlight the advantages of field-aligned interpolation. The same methodology is successfully applied to the solution of the gyrokinetic Vlasov equation, for which we present the ion temperature gradient (ITG) instability as a classical test-case: first we solve this in cylindrical geometry (screw-pinch), and next in toroidal geometry (circular Tokamak). In the first case, the algorithm is implemented in Selalib (semi-Lagrangian library), and the numerical simulations provide linear growth rates that are in accordance with the linear dispersion analysis. In the second case, the algorithm is implemented in the Gysela code, and the numerical simulations are benchmarked with those employing the standard (not aligned) scheme. Numerical experiments show that field-aligned interpolation leads to considerable memory savings for the same level of accuracy; substantial savings are also expected in reactor-scale simulations. (authors)

[en] Within the framework of researches on controlled nuclear fusion, and more particularly on the understanding of magneto-hydrodynamic stability and of plasma control, the contributions of this publication are more or less brief scientific articles which address the following related topics: Magneto-hydrodynamic stability; Magnetic reconnection; Magnetic islands; Island and turbulence; Saw-teeth within plasma; Stability of plasmas in combustion; Relaxations at the edge of plasma; At the frontier of the operational domain; Detecting the invisible; Useful notions about the confinement of a plasma by a magnetic field