The problem of multidimensional radiative transfer has been an immensely challenging one in the general areas of combustion, controlled thermonuclear fusion and computational astrophysics. This is so despite the fact that large supercomputers at NSF-funded centers have enough memory to solve modest sized radiative transfer problems and supercomputers within DOE have enough memory to solve large sized problems. Several factors other than computer memory pose a problem. First, many of the algorithms for radiative transfer have low (first) order of accuracy thus resulting in confusion between discretization error and optical depth effects. Second, some of the popular higher order methods converge rather slowly. Third, some of the algorithms result in partial serialization of the solution strategy, thus nullifying the advantages of a massively parallel supercomputer. Fourth, parallelism and adaptivity are hard to achieve simultaneously in this context.
In this work we focus on S_n techniques for multidimensional radiative transfer. Several higher order discretization strategies are examined for their accuracy and rate of convergence. Superior methods are found which permit convergence to better than discretization error in very few operations. The solution strategy is inherently parallel and does not suffer from the problem of having to make ordered sweeps which is the cause of the above-mentioned partial serialization. Furthermore, the solution strategy takes well to AMR techniques resulting in a parallel, self-adaptive strategy for multidimensional radiative transfer.