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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 thisapproach the determination of $\sigma(u)$ is broken in two pieces: a change ofthe flow time at fixed physical size, and a change of the size of the system atfixed flow time. Using both perturbative arguments and a set of simulations inthe pure gauge theory we show that this approach leads to a better control overthe continuum extrapolations. Following this new proposal we determine therunning coupling at high energies in the pure gauge theory and re-examine thedetermination of the $\Lambda$-parameter, with special care on the perturbativetruncation uncertainties.

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