Expansion of a Strömgren sphere in 1D using A1 with an ambient gas density of n_H = 10   cm^-3 (results for higher densities are almost indistinguishable). The position of the cell with the steepest radial gradient in H⁺ fraction is plotted as a function of time for simulations with cell optical depth Δτ₀ ≃ 1 and dt00-dt04 (top left), Δτ₀ ≃ 3 and dt00-dt04 (top right), Δτ₀ ≃ 10 and dt00-dt04 (centre left), Δτ₀ ≃ 30 and dt00-dt04 (centre right), Δτ₀ ≃ 10 and dt05-dt08 (below left), and Δτ₀ ≃ 30 and dt05-dt08 (below right). The equivalent plots for dt09-dt12 all show results indistinguishable from the dt12 curve. The trend for decreasing accuracy with increasing cell optical depth (i.e. decreasing numerical resolution) for criteria dt00-dt04 is clearly seen in the first four panels; this is corrected by the criteria dt05-dt08 that also limit the timestep by the relative change in internal energy. Discreteness in the computational grid and the output frequency account for the non-smooth curves in the plots for Δτ₀ ≃ 10 and 30 (also in the following figures).