Radiative transfer (RT) simulations are now at the forefront of numerical astrophysics. They are becoming crucial for an increasing number of astrophysical and cosmological problems; at the same time their computational cost has come within reach of currently available computational power. Further progress is retarded by the considerable number of different algorithms (including various flavours of ray tracing and moment schemes) developed, which makes the selection of the most suitable technique for a given problem a non‐trivial task. Assessing the validity ranges, accuracy and performances of these schemes is the main aim of this paper, for which we have compared 11 independent RT codes on five test problems: (0) basic physics; (1) isothermal H ii region expansion; (2) H ii region expansion with evolving temperature; (3) I‐front trapping and shadowing by a dense clump and (4) multiple sources in a cosmological density field. The outputs of these tests have been compared and differences analysed. The agreement between the various codes is satisfactory although not perfect. The main source of discrepancy appears to reside in the multifrequency treatment approach, resulting in different thicknesses of the ionized‐neutral transition regions and the temperature structure. The present results and tests represent the most complete benchmark available for the development of new codes and improvement of existing ones. To further this aim all test inputs and outputs are made publicly available in digital form.

In this paper, we present a method for the solution of those linear transport processes that may be described by a master equation, such as electron, neutron, and photon transport, and more exotic variants thereof. We base our algorithm on a Markov process on a Voronoi-Delaunay grid, a nonperiodic lattice which is derived from a random point process that is chosen to optimally represent certain properties of the medium through which the transport occurs. Our grid is locally translation and rotation invariant in the mean. We illustrate our approach by means of a particular example, in which the expectation value of the length of a grid line corresponds to the local mean free path. In this example, the lattice is a direct representation of the "free path space" of the medium. Subsequently, transport is defined as simply moving particles from one node to the next, interactions taking place at each point. We derive the statistical properties of such lattices, describe the limiting behavior, and show how interactions are incorporated as global coefficients. Two elementary linear transport problems are discussed: that of free ballistic transport, and the transport of particles through a scattering medium. We also mention a combination of these two. We discuss the efficiency of our method, showing that it is much faster than most other methods because the operation count does not scale with the number of sources. We test our method by focusing on the transport of ionizing radiation through a static medium, and show that the computed results for the classical test case of an ionization front expanding in a homogeneous medium agree perfectly with the analytic solution. We finish by illustrating the efficiency and flexibility of our method with the results of a simulation of the reionization of the large scale structure of the Universe.

Abstract. In this Letter, we argue that the standard analytical derivations of properties of HII regions, such as the speed, shape and asymptotic position of ionisation fronts require a more precise treatment. These derivations use the on the spot approximation, which in effect ignores the diffuse component of the radiation field. We show that, in fact, HII regions are diffusion dominated. This has as a result that the morphology of inhomogeneous HII regions will be drastically different, because shadowing effects have a less profound impact on the apparent shape. Moreover, it will have influence on the propagation speed of ionisation fronts. We quantify our claims by analytically deriving the internal radiation structure of HII regions, taking diffusion fully into account for several different cosmologically relevant density distributions.

We present a new radiative transfer method, the SimpleX method, that solves the radiative transfer equation on an unstructured, physical grid. SimpleX is a computationally very fast method that does not scale with the number of sources in the simulation domain. Tests show that the results of ionization front calculations done with SimpleX are comparable to the results of other methods, but at lower computational costs. It is important to incorporate diffuse photons in calculations of ionization fronts. The outer parts of H II regions are dominated by diffuse photons, so we need to incorporate these to model the shape and dynamics of ionization fronts properly. Because SimpleX does not scale with the number of sources in the simulation domain, we can model diffuse photons without extra computational costs. Our goal is to couple SimpleX to the hydrodynamics code FLASH to calculate radiative transfer in step with the hydrodynamics calculations. We plan to investigate primordial star formation, specifically the interaction of radiation with the accreting protostar, to determine the final mass of the first stars, and the influence of the star's UV radiation on the surrounding minihalo.