  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) $E\times B$ 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.
