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

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