Asymptotic Derivation and Numerical Investigation of Time-Dependent Simplified Pn Equations

TL;DR

This paper derives and investigates time-dependent simplified Pn equations up to order 3, demonstrating their improved accuracy and efficiency over diffusion and higher-order Pn models through numerical simulations.

Contribution

The paper provides an asymptotic derivation of time-dependent simplified Pn equations up to n=3 and explores their properties and numerical performance.

Findings

SPn equations are hyperbolic and differ from previous models.

Numerical results show SPn equations are more accurate than diffusion.

For low n, SPn models are more efficient than comparable Pn models.

Abstract

The steady-state simplified Pn (SPn) 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 Pn equations up to n = 3. Additionally, SPn equations of arbitrary order are derived in an ad hoc way. The resulting SPn 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 Pn and SPn equations are performed. We simulate neutron distributions of a moving rod and present results for a benchmark problem, known as the checkerboard problem. The SPn 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 Pn models of comparable cost.