A. Ishizawa

H. Tsutsui

N. Nakajima

S. Maeyama

S. Tsuji-Iio

T.-H. Watanabe

National Institute for Fusion Science (NIFS-Repository)

A semi-Lagrangian scheme is applied for the ﬁrsttime to computations of charged particle motions alongmagnetic ﬁeld lines, to numerically solve the δf gyroki-netic equations in a ﬂux tube geometry. This new solveradopted in the gyrokinetic Vlasov simulations has anadvantage over the conventional Eulerian codes in cal-culating the parallel dynamics, because semi-Lagrangianschemes are free of the Courant-Friedrichs-Lewy (CFL)condition that restricts the time step size. A study of theaccuracy of the parallel motion simulations reveals thatnumerical errors mainly stem from spatial (not tempo-ral) discretization for realistic values of the grid spacingand time step, and it demonstrates the advantage of thesemi-Lagrangian scheme. This novel numerical methodis successfully applied to linear gyrokinetic simulationsof the ion temperature gradient instability, where timesteps larger than those restricted by the CFL conditioncan be employed 1).The parallel dynamics of the perturbed ion distri-bution function f in the gyrokinetic Vlasov equation isgiven by∂f∂t+ {f,K}∥ = 0, (1)where K = v2∥/2 + µB/mi is the particle kinetic en-ergy per unit mass and {f, g}∥ = ∇∥f∂v∥g − ∂v∥f∇∥gdenotes the parallel Poisson brackets. Equation (1) ex-presses that the parallel dynamics is regarded as an ad-vection of a distribution function along equi-contour linesof the particle kinetic energy K in the parallel phasespace (z, v∥) shown in Fig. 1 (a). The contours withK < µB0/mi correspond to trapped particle trajecto-ries, and the others represent trajectories of passing par-ticles. Time integration of Eq. (1) is successfully com-puted by means of a semi-Lagrangian scheme with thesecond-order operator splitting method. The snapshotsFig. 2: Linear growth rates γl and real frequencies ωras a function of the time step size ∆t. Results using thesemi-Lagrangian scheme for the parallel dynamics arecompared with the original GKV results.of equi-contours of the distribution function are shown inFig. 1 (b)-(d). The ballistic motion of passing particleselongates the proﬁle and makes ﬁne-scale structures.By implementing the semi-Lagrangian scheme tothe GKV code 2), tests of a gyrokinetic simulation ofthe linear ITG instability are carried out for the CycloneDIII-D base case parameter set. Figure 2 plots the lin-ear growth rate and real frequency as a function of thetime step size for a low-ky mode, where the CFL restric-tion due to the parallel motions is more severe than thatdue to the perpendicular drift motions in the originalEulerian GKV code. The results agree well with thoseobtained by the original GKV code: their relative errorsare less than 0.1%. Therefore, the new solver with thesemi-Lagrangian scheme enables us to take larger timestep sizes than the CFL restriction and still achieve ac-curate results.1) Maeyama, S., Ishizawa, A., Watanabe, T.-H., Naka-jima, N., Tsuji-Iio, S., Tsutsui, H. : Plasma and Fu-sion Research 6, (2011) to be published.2) Watanabe, T.-H., Sugama, H., : Nuclear Fusion 46,24 (2006).Fig. 1: (a) Contour lines of the particle kinetic energy K, and (b)-(d) snapshots of equi-contours of the distributionfunction f in parallel phase space. Horizontal and vertical axes are deﬁned by the ﬁeld-aligned coordinate z and theparallel velocity v∥, respectively.358§19. A Numerical Method for Parallel Particle Motions in Gyrokinetic Vlasov SimulationsMaeyama, S., Tsuji-Iio, S., Tsutsui, H. (Tokyo Institute of Technology),  Ishizawa, A., Watanabe, T.-H., Nakajima, N.

A Numerical Method for Parallel Particle Motions in Gyrokinetic Vlasov Simulations

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