A. Ishizawa

H. Tsutsui

N. Nakajima

S. Maeyama

S. Tsuji-Iio

T.-H. Watanabe

National Institute for Fusion Science (NIFS-Repository)

A hybrid method of semi-Lagrangian and additivesemi-implicit Runge-Kutta schemes is developed for gy-rokinetic Vlasov simulations in a ﬂux tube geometry.The time-integration scheme is free from the Courant-Friedrichs-Lewy condition for the linear advection termsin the gyrokinetic equation. The new method is appliedto simulations of the ion-temperature-gradient instabil-ity in fusion plasmas conﬁned by helical magnetic ﬁelds,where the parallel advection term severely restricts thetime step size for explicit Eulerian schemes. Linear andnonlinear results show good agreements with those ob-tained by using the explicit Runge-Kutta-Gill scheme,while the new method substantially reduces the compu-tational cost 1).In order to overcome time-step-size restrictionsfrom linear terms in the gyrokinetic equations, we ap-ply a semi-Lagrangian scheme to the parallel advec-tion and the mirror force terms, and deal with themagnetic drift term implicitly by the use of an opera-tor splitting scheme. Implementing the hybrid methodof semi-Lagrangian and additive semi-implicit Runge-Kutta schemes (SLASIRK) to the GKV code, we carriedout linear and nonlinear simulations of the ITG insta-bility in helical plasmas. Figure 1 plots the norm of theerrors of the eigenfunction LE, the errors of the linear10-610-510-410-310-210-1100 0.01  0.1  1Errors L E, Lγ, L ωTime step size Δt vti/Ln∝ Δt2LELγLωFig. 1: The errors of the eigen function LE, growth rateLγ and real frequency Lω plotted as a function of thetime step size Δt are shown by circle, square and triangledots, respectively (where kyρti = 0.417 and SLASIRK isemployed). The solid line represents a reference slope forthe second-order accuracy.growth rate Lγ and the real frequency Lω for the lin-early most unstable mode as a function of the time stepsize. The results of LE shows SLASIRK has the second-order temporal accuracy as expected, while Lγ and Lωare more quickly converged. The norm of errors tends toapproach a certain constant value for Δt < 0.02Ln/vti,where spatial errors dominate temporal ones. We notethat explicit Eulerian schemes require time steps smallerthan 0.005Ln/vti for stable computations.Figure 2 shows elapsed time for a hundred simu-lation steps by employing the forth-order Runge-Kutta-Gill scheme (RKG) and SLASIRK. The amount of com-putations involved in the second-order SLASIRK is lessthan that in the fourth-order RKG, since the former com-putes the Poisson solver and nonlinear term twice pertime step and the latter does four times. Excluding datainput and output, RKG and SLASIRK take 7.79s and4.45s per time step, respectively.The newly-developed method allows twenty (three)times larger time step size for linear (nonlinear) runsthan the original GKV which employs the forth-order ex-plicit Runge-Kutta-Gill method with enough accuracy.By combining a large time step size and short elapsedtime per time step, the eﬃciency of the gyrokineticVlasov simulation is largely enhanced by utilizing thepresent method.1) Maeyama, S., Ishizawa, A., Watanabe, T.-H., Naka-jima, N., Tsuji-Iio, S., Tsutsui, H. : ComputerPhysics Communications 183, (2012) in press. 0 100 200 300 400 500 600 700 800 900RKG SLASIRKElapsed time for 100 simulation steps [s]PoissonRKLinear termsExB termSLOthersFig. 2: Comparison of elapsed time between the fourth-order RKG and the second-order SLASIRK for a hun-dred simulation steps. From the bottom, boxes representelapsed time to compute the Poisson solver, time integra-tion by RKG or ASIRK, linear terms, nonlinear E ×Bterm, time integration by semi-Lagrangian schemes, andthe others including data input and output.351§10. Hybrid Method of Semi-Lagrangian and Additive Semi-implicit Runge-Kutta  Schemes for Gyrokinetic Vlasov SimulationsMaeyama, S., Tsuji-Iio, S., Tsutsui, H. (Tokyo Institute of Technology),  Ishizawa, A., Watanabe, T.-H., Nakajima, N.

Hybrid Method of Semi-Lagrangian and Additive Semi-implicit Runge-Kutta Schemes for Gyrokinetic Vlasov Simulations

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