Second-order neutron transport methods

Among the approaches to obtaining numerical solutions for neutral particle transport problems, those classified as second-order or even-parity methods have found increased use in recent decades. First-order and second-order methods differ in a number of respects. Following discretization of the energy variable, invariably through some form of the multigroup approximation, the time-independent forms of both are differential in the spatial variable and integral in angle. They differ in that the more conventional first-order equation includes only first derivatives in the spatial variables, but requires solution over the entire angular domain. Conversely, the second-order form includes second derivatives but requires solution over one half of the angular domain. The two forms in turn lead to contrasting approaches to reducing the differential-integral equations to sets of linear equations and in the formulation of iterative methods suitable for the numerical solution of large engineering design problems. In what follows, we explore the state of methods used to solve the second-order transport equation, comparing them, where possible, to first-order methods.