We present nested-grid, high-resolution hydrodynamic simulations of gas and particle dynamics in the vicinity of Mars- to Earth-mass planetary embryos. The simulations extend from the surface of the embryos to a few vertical disc scale heights, with a spatial dynamic range of ∼1.4 × 105. Our results confirm that ‘pebble’-sized particles are readily accreted, with accretion rates continuing to increase up to metre-size ‘boulders’ for a 10 per cent MMSN surface density model. The gas mass flux in and out of the Hill sphere is consistent with the Hill rate, R2 H = 4 10−3 M⊕ yr−1. While smaller size particles mainly track the gas, a net accretion rate of ≈2 10−5 M⊕ yr−1 is reached for 0.3–1 cm particles, even though a significant fraction leaves the Hill sphere again. Effectively, all pebble-sized particles that cross the Bondi sphere are accreted. The resolution of these simulations is sufficient to resolve accretion-driven convection. Convection driven by a nominal accretion rate of 10−6 M⊕ yr−1 does not significantly alter the pebble accretion rate. We find that, due to cancellation effects, accretion rates of pebble-sized particles are nearly independent of disc surface density. As a result, we can estimate accurate growth times for specified particle sizes. For 0.3–1 cm size particles, the growth time from a small seed is ∼0.15 million years for an Earth-mass planet at 1 au and ∼0.1 million years for a Mars mass planet at 1.5 au.