Mathematical properties of the SimpleX algorithm

Context. IntheSimpleX radiative transfer algorithm, photons are transported on an unstructured Delaunay triangulation. This approach is non-standard, requiring a thorough analysis of possible systematic effects. Aims. We verify whether the SimpleX radiative transfer algorithm conforms to mathematical expectations and develop both an error analysis and improvements to earlier versions of the code. Methods. We use numerical simulations and classical statistics to obtain quantitative descriptions of the systematics of the SimpleX algorithm. Results. We present a quantitative description of the error properties of SimpleX, numerical validation of the method and verification of the analytical results. Furthermore we describe improvements in accuracy and speed of the method. Conclusions. It is possible to transport particles such as photons in a physically correct manner with the SimpleX algorithm. This requires the use of weighting schemes or the modification of the point process underlying the transport grid. We explore and apply several possibilities.