A New Fast Monte Carlo Code for Solving Radiative Transfer Equations based on Neumann Solution

Abstract: In this paper, we proposed a new Monte Carlo radiative transport (MCRT) scheme, which is based completely on the Neumann series solution of Fredholm integral equation. This scheme indicates that the essence of MCRT is the calculation of infinite terms of multiple integrals in Neumann solution simultaneously. Under this perspective we redescribed MCRT procedure systematically, in which the main work amounts to choose an associated probability distribution function (PDF) for a set of random variables and the corresponding unbiased estimation functions. We can select a relatively optimal estimation procedure that has a lower variance from an infinite possible choices, such as the term by term estimation. In this scheme, MCRT can be regarded as a pure problem of integral evaluation, rather than as the tracing of random walking photons. Keeping this in mind, one can avert some subtle intuitive mistakes. In addition the $\delta$-functions in these integrals can be eliminated in advance by integrating them out directly. This fact together with the optimal chosen random variables can remarkably improve the Monte Carlo (MC) computational efficiency and accuracy, especially in systems with axial or spherical symmetry. An MCRT code, Lemon (Linear Integral Equations' Monte Carlo Solver Based on the Neumann solution), has been developed completely based on this scheme. Finally, we intend to verify the validation of Lemon, a suite of test problems mainly restricted to flat spacetime have been reproduced and the corresponding results are illustrated in detail.