Efficient integration of gradient flow in lattice gauge theory and properties of low-storage commutator-free Lie group methods

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.