Asymptotic derivation and numerical investigation of time-dependent simplified equations

The steady-state simplified P N ( SP N ) approximations to the linear Boltzmann equation have been proven to be asymptotically higher-order corrections to the diffusion equation in certain physical systems. In this paper, we present an asymptotic analysis for the time-dependent simplified P N equations up to N = 3 . Additionally, SP N equations of arbitrary order are derived in an ad hoc way. The resulting SP N equations are hyperbolic and differ from those investigated in a previous work by some of the authors. In two space dimensions, numerical calculations for the P N and SP N equations are performed. We simulate neutron distributions of a moving rod and present results for a benchmark problem, known as the checkerboard problem. The SP N equations are demonstrated to yield significantly more accurate results than diffusion approximations. In addition, for sufficiently low values of N, they are shown to be more efficient than P N models of comparable cost.