  The smoothing procedure known as the gradient flow that suppresses
ultraviolet fluctuations of gauge fields plays an important role in lattice
gauge theory calculations. In particular, this procedure is often used for
high-precision scale setting and renormalization of operators. The gradient
flow equation is defined on the SU(3) manifold and therefore requires
geometric, or structure-preserving, integration methods to obtain its numerical
solutions. We examine the properties and origins of the three-stage third-order
explicit Runge-Kutta Lie group integrator commonly used in the lattice gauge
theory community, demonstrate its relation to 2N-storage classical Runge-Kutta
methods and explore how its coefficients can be tuned for optimal performance
in integrating the gradient flow. We also compare the performance of the tuned
method with two third-order variable step size methods. Next, based on the
recently established connection between low-storage Lie group integrators and
classical 2N-storage Runge-Kutta methods, we study two fourth-order low-storage
methods that provide a computationally efficient alternative to the commonly
used third-order method while retaining the convenient iterative property of
the latter. Finally, we demonstrate that almost no coding effort is needed to
implement the low-storage Lie group methods into existing gradient flow codes.
