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Following a recent paper by Fodor et al. (arXiv:1406.0827), we reexamineseveral types of tree-level improvements on the flow action with various gaugeactions in order to reduce the lattice discretization errors in the Yang-Millsgradient flow method. We propose two types of tree-level, $\mathcal{O}(a^4)$improved lattice gradient flow including the rectangle term in both the flowand gauge action within the minimal way. We then perform numerical simulationswith the simple plaquette gauge action for testing our proposal. Our numericalresults of the expectation value of the action density, $\langle E(t)\rangle$,show that two $\mathcal{O}(a^4)$ improved flows significantly eliminate thediscretization corrections in the small flow time $t$ regime. On the otherhand, the values of $t^2\langle E(t)\rangle$ in the large $t$ regime, where thelattice spacing dependence of the tree-level term dies out as inverse powers of$t/a^2$, are different between the results given by two optimal flows leadingto the same $\mathcal{O}(a^4)$ improvement at tree level. This may suggest thatnon-negligible $\mathcal{O}(g^2 a^2)$ effect sets in the large $t$ regime,where the running coupling $g(1/\sqrt{8t})$ becomes large.

Lattice gradient flow with tree-level $\mathcal{O}(a^4)$ improvement in pure Yang-Mills theory