An analysis of systematic effects in finite size scaling studies using the gradient flow

We propose a new strategy for the determination of the step scaling function $\sigma(u)$ in finite size scaling studies using the Gradient Flow. In this approach the determination of $\sigma(u)$ is broken in two pieces: a change of the flow time at fixed physical size, and a change of the size of the system at fixed flow time. Using both perturbative arguments and a set of simulations in the pure gauge theory we show that this approach leads to a better control over the continuum extrapolations. Following this new proposal we determine the running coupling at high energies in the pure gauge theory and re-examine the determination of the $\Lambda$-parameter, with special care on the perturbative truncation uncertainties.