A mixed P1–DP0 diffusion theory for planar geometry

A variational analysis is used to derive a mixed P1–DP0 (P1 spherical harmonics–double P0 spherical harmonics) angular approximation to the time-independent monoenergetic neutron transport equation in one-dimensional planar geometry. This mixed angular approximation contains a space-dependent weight factor α(x) that controls the local angular approximation used at a spatial point x: α(x) = 1 yields the standard P1 (diffusion) approximation, α(x) = 0 gives the standard DP0 approximation, and 0 < α(x) < 1 produces a mixed P1–DP0 angular approximation. The diffusion equation obtained differs from the standard P1 diffusion equation only in the definition of the diffusion coefficient. Standard Marshak incident angular flux boundary conditions are also obtained via the variational analysis. We examine the use of this mixed angular approximation coupled with the standard P1 approximation to more accurately treat material interfaces and vacuum boundaries. We propose a simple but effective functional form for the weight factor α(x) that removes the need for the user to specify the value. Numerical results from several test problems are presented to demonstrate that significant improvements in accuracy can be obtained using this method with essentially no computational penalty.