Expansion of a Strömgren sphere in 1D for the three algorithms: the implicit A1 (top), first-order explicit A2 (centre), and second-order A3 (bottom), with different timestep criteria as indicated. The ratio of actual to theoretical ionisation front radius is plotted on a logarithmic time axis from t = 0.001t_rec for A2 and A3, and from 0.1t_rec for A1 (because the timestep criterion is much less restrictive for A1). The different convergence rates for A2 and A3 are again apparent.