A hybrid method of semi-Lagrangian and additive semi-implicit Runge-Kutta schemes for gyrokinetic Vlasov simulations

Abstract A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes is developed for gyrokinetic Vlasov simulations in a flux tube geometry. The time-integration scheme is free from the Courant–Friedrichs–Lewy condition for the linear advection terms in the gyrokinetic equation. The new method is applied to simulations of the ion-temperature-gradient instability in fusion plasmas confined by helical magnetic fields, where the parallel advection term severely restricts the time step size for explicit Eulerian schemes. Linear and nonlinear results show good agreements with those obtained by using the explicit Runge–Kutta–Gill scheme, while the new method substantially reduces the computational cost.

[1]  Shunji Tsuji-Iio,et al.  A Numerical Method for Parallel Particle Motions in Gyrokinetic Vlasov Simulations , 2011 .

[2]  Gregory W. Hammett,et al.  Field‐aligned coordinates for nonlinear simulations of tokamak turbulence , 1995 .

[3]  A. Lenard,et al.  PLASMA OSCILLATIONS WITH DIFFUSION IN VELOCITY SPACE , 1958 .

[4]  H. Sugama,et al.  Reduction of turbulent transport with zonal flows enhanced in helical systems. , 2008, Physical review letters.

[5]  Laurent Villard,et al.  Flux- and gradient-driven global gyrokinetic simulation of tokamak turbulence , 2011 .

[6]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[7]  Frank Jenko,et al.  Electron temperature gradient driven turbulence , 1999 .

[8]  Alain J. Brizard,et al.  Foundations of Nonlinear Gyrokinetic Theory , 2007 .

[9]  Laurent Villard,et al.  Gyrokinetic simulations of turbulent transport , 2010 .

[10]  Hideo Sugama,et al.  Study of electromagnetic microinstabilities in helical systems with the stellarator expansion method , 2004 .

[11]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[12]  Hideo Sugama,et al.  Zonal flows and ion temperature gradient instabilities in multiple-helicity magnetic fields , 2007 .

[13]  G. Knorr,et al.  The integration of the vlasov equation in configuration space , 1976 .

[14]  X. Garbet,et al.  Global full-f gyrokinetic simulations of plasma turbulence , 2007 .

[15]  R. Waltz,et al.  Anomalous transport scaling in the DIII-D tokamak matched by supercomputer simulation. , 2003, Physical review letters.

[16]  Shinji Tokuda,et al.  Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation , 2008, Comput. Phys. Commun..

[17]  Xiaolin Zhong,et al.  Additive Semi-Implicit Runge-Kutta Methods for Computing High-Speed Nonequilibrium Reactive Flows , 1996 .

[18]  F. Jenko,et al.  Gyrokinetic simulation of collisionless trapped-electron mode turbulence , 2005 .

[19]  E. Frieman,et al.  Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria , 1981 .

[20]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[21]  H. Yoshida.  Construction of higher order symplectic integrators , 1990 .

[22]  F. Jenko,et al.  Electron temperature gradient turbulence. , 2000, Physical review letters.

[23]  R. E. Waltz,et al.  An Eulerian gyrokinetic-Maxwell solver , 2003 .

[24]  Mike Kotschenreuther,et al.  Comparison of initial value and eigenvalue codes for kinetic toroidal plasma instabilities , 1995 .

[25]  Hideo Sugama,et al.  Velocity–space structures of distribution function in toroidal ion temperature gradient turbulence , 2011 .

[26]  M. Wakatani,et al.  Stellarator and heliotron devices , 1998 .

[27]  F. J. Casson,et al.  The nonlinear gyro-kinetic flux tube code GKW , 2009, Comput. Phys. Commun..