Towards optimal explicit time-stepping schemes for the gyrokinetic equations

Abstract

The nonlinear gyrokinetic equations describe plasma turbulence in laboratory and astrophysical plasmas. To solve these equations, massively parallel codes have been developed and run on present-day supercomputers. This paper describes measures to improve the efficiency of such computations, thereby making them more realistic. Explicit Runge-Kutta schemes are considered to be well suited for time-stepping. Although the numerical algorithms are often highly optimized, performance can still be improved by a suitable choice of the time-stepping scheme, based on spectral analysis of the underlying operator. Here, an operator splitting technique is introduced to combine first-order Runge-Kutta-Chebychev schemes for the collision term with fourth-order schemes for the remaining terms. In the nonlinear regime, based on the observation of eigenvalue shifts due to the (generalized) E × B advection term, an accurate and robust estimate for the nonlinear timestep is developed. The presented techniques can reduce simulation times by factors of up to three in realistic cases. This substantial speedup encourages the use of similar timestep optimized explicit schemes not only for the gyrokinetic equation, but also for other applications with comparable properties.

Global gyrokinetic simulations of ITG turbulence in the magnetic configuration space of the Wendelstein 7-X stellarator

“…This allows one to reduce the original hyperbolic integro-differential system of equations to a system of ordinary differential equations, which are then explicitly integrated in time. GENE-3D currently uses a 4th order explicit Runge-Kutta (RK4) integrator; the timestep is computed at the initialization and maximized during a run to ensure optimal stability [24]. Because of spatial dependencies in all three spatial coordinates, fourth-order centered finite difference schemes are used to calculate derivatives.…”

An exponential integrator for the drift-kinetic model

“…In Fig. 3, we plot the error of the total mass defined in (9) and compare our method with the results from the semi-Lagrangian method. We see that the mass in both the perturbation method as well as the direct formulation is much better preserved than for the semi-Lagrangian approach.…”

“…Some high-order numerical methods that avoid splitting have also been proposed for gyrokinetic type models. The time integration is mostly based on explicit Runge-Kutta methods (see [3,9]) which suffer from a CFL condition.…”