cons.

When radiation sources are embedded in media on meshes, RT calculations can be categorised into two types; one premises that the source functions are assigned on meshes and the other does that radiation sources are treated as point sources independent of meshes. In the former type, the RT equations are integrated along long or short characteristics between meshes. The latter is advantageous when the number of the point sources, $N_{\rm s}$, is smaller than that of the boundary meshes, $\sim N_{\rm m}^{2/3}$, where $N_{\rm m}$ is the total number of the meshes. The latter type of the RT schemes is often called ‘ray-tracing’ that we deal with in this paper.

The most accurate and straight-forward RT scheme is the long characteristics method in which all source meshes are connected to all other relevant meshes [@abel99; @sokasian01; @RSPH]. This method is however very expensive computationally. The computational costs scales with $N_{\rm m}^2$ in general and with $N_{\rm m}^{4/3} N_{\rm s}$ for the transfer from point sources.

The short characteristics method [@kunasz88; @stone92; @mellema98; @nakamoto01] reduces the computational cost by integrating the equation of RT only along lines that connect nearby cells. It scales with $N_{\rm m}^{5/3}$ and with $N_{\rm m} 