Transport Error Bounds Via P_N Approximations

By using variational means, it is found for any one velocity system with non- zero absorption cross section having either vacuum, reflecting, or antireflecting boundary conditions that the transport solution is, in a very specific sense, approached monotonically from above by the solutions to the odd P_N equations and from below by the solutions to the even P_N equations, provided the P_N solutions are obtained by using appropriate continuity and external boundary conditions. That is to say, odd and even P_N calculations “bracket” the transport solution. In one instance, the escape probability is bounded and, in another, the disadvantage factor. This theoretical result, along with certain numerical evidence, suggests that the modified P₂ approximation of Dawson may serve as a practical, reasonably accurate alternative to diffusion theory for certain realistic design problems.