doi:10.1016/j.cpc.2004.07.009
Copyright © 2004 Elsevier B.V. All rights reserved.
A semi-Lagrangian code for nonlinear global simulations of electrostatic drift-kinetic ITG modes
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M. Brunettia, 
, 
, V. Grandgirardb, O. Sautera, J. Vaclavika and L. Villarda
aCentre de Recherches en Physique des Plasmas, Association EURATOM – Confédération Suisse, EPFL, 1015 Lausanne, Switzerland
bDRFC Association Euratom-CEA, CEA Cadarache, 19108 St Paul-lez-Durance, France
Received 20 April 2004;
accepted 6 July 2004.
Available online 16 September 2004.
Abstract
A semi-Lagrangian code for the solution of the electrostatic drift-kinetic equations in straight cylinder configuration is presented. The code, CYGNE, is part of a project with the long term aim of studying microturbulence in fusion devices. The code has been constructed in such a way as to preserve a good control of the constants of motion, possessed by the drift-kinetic equations, until the nonlinear saturation of the ion-temperature-gradient modes occurs. Studies of convergence with phase space resolution and time-step are presented and discussed. The code is benchmarked against electrostatic Particle-in-Cell codes.
Keywords: Semi-Lagrangian; Electrostatic; Ion-temperature-gradient; Drift-kinetic; Conservation laws
PACS: 52.65; 52.35.Q; 52.25.F; 52.55
Fig. 1. Time evolution of the relative error in the entropy (a) and the L2-norm (b) without (solid line) and with (dashed line) logarithmic interpolation for the case (Nr,Nθ,Nz,Nv
)=(64,64,64,64) and ΔtΩi=0.2.
Fig. 2. ITG growthrates (a) and frequencies (b): values obtained with CYGNE (×) and LORB5 (o).
Fig. 3. Time evolution of the (perturbed) field energy obtained with CYGNE (solid line) and ORB5 (dashed line).
Fig. 4. Time evolution of the (perturbed) field and kinetic energy, and of the radial heat flux for different phase space resolutions: (Nr,Nθ,Nz,Nv)=(64,256,64,32) in solid line, (128,256,32,32) in dashed line, (32,128,32,32) in dotted–dashed line, (32,128,32,32) with Newton–Raphson algorithm in dotted line, and (32,128,32,32) using equidistant meshes in dotted line and ‘+’ markers.
Fig. 5. Time evolution of the relative error in the number of ions (frame (a)), the total energy (b), the entropy (c) and the L2-norm (d) for different phase space resolutions: (Nr,Nθ,Nz,Nv)=(64,256,64,32) in solid line, (128,256,32,32) in dashed line, (32,128,32,32) in dotted–dashed line, (32,128,32,32) with Newton–Raphson algorithm in dotted line, and (32,128,32,32) using equidistant meshes in dotted line and ‘+’ markers.
Fig. 6. Time evolution of the (perturbed) field and kinetic energy, and of the radial heat flux for the same phase space resolution, (Nr,Nθ,Nz,Nv)=(64,256,64,32), and different time-steps: ΔtΩi=0.2 (solid line), ΔtΩi=0.5 (dashed line) and ΔtΩi=1 (dotted–dashed line).
Fig. 7. Time evolution of the relative error in the number of ions (frame (a)), the total energy (b), the entropy (c) and the L2-norm (d) for the same resolution, (Nr,Nθ,Nz,Nv)=(64,256,64,32), and different time-steps: ΔtΩi=0.2 (solid line), ΔtΩi=0.5 (dashed line) and ΔtΩi=1 (dotted–dashed line).
Fig. 8. Cross-section of the distribution function in the r–θ plane for v=0 and z=L/3 at time tΩi=1200.
Table 1.
CPU time (normalized to the time required in the case of Bulirsch–Stoer (BS) and splitting sequence S2 given by Eq. (15)) for a grid (Nr,Nθ,Nz,Nv)=(64,256,64,32). NR means ‘Newton–Raphson’ and the splitting sequence S1 is given by Eq. (14)
Table 2.
CPU time (normalized to the time required in the case with non-equidistant meshes) and memory requirements per processor (the last column refers to the memory size during communications) with and without equidistant meshes (EM) in r–v directions, using Nproc=4
Table 3.
Parameter set for the comparison case
Table A.1
Values of the spline basis Bi,4, Bi,3 and 
at given points
Corresponding author. Tel.: +41-(0)21-693 65 29; fax: +41-(0)21-693 51 76.
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