Accuracy and Efficiency of Raytracing Photoionisation Algorithms
  Three non-equilibrium photoionisation algorithms for hydrodynamical
grid-based simulation codes are compared in terms of accuracy, timestepping
criteria, and parallel scaling. Explicit methods with first-order time accuracy
for photon conservation must use very restrictive timestep criteria to
accurately track R-type ionisation fronts. A second-order accurate algorithm is
described which, although it requires more work per step, allows much longer
timesteps and is consequently more efficient. Implicit methods allow ionisation
fronts to cross many grid cells per timestep while maintaining photon
conservation accuracy. It is shown, however, that errors are much larger for
multi-frequency radiation than for monochromatic radiation with the implicit
algorithm used here, and large errors accrue when an ionisation front crosses
many optical depths in a single step. The accuracy and convergence rates of the
different algorithms are tested with a large number of timestepping criteria to
identify the best criterion for each algorithm. With these criteria selected,
the second-order explicit algorithm is the most efficient of the three, and its
parallel scaling is significantly better than that of the implicit algorithm.
The upgrade from first- to second-order accuracy in explicit algorithms could
be made very simply to fixed-grid and adaptive mesh-refinement codes which
currently use a first-order method.
