The Time-Dependent Simplified P₂ Equations: Asymptotic Analyses and Numerical Experiments

Using an asymptotic expansion, we found that the modified time-dependent simplified P₂ (SP₂) equations are robust, high-order, asymptotic approximations to the time-dependent transport equation in a physical regime in which the conventional time-dependent diffusion equation is the leading-order approximation. Using diffusion limit analysis, we also asymptotically compared three competitive time-dependent equations (the telegrapher's equation, the time-dependent SP₂ equations, and the time-dependent simplified even-parity equation). As a result, we found that the time-dependent SP₂ equations contain higher-order asymptotic approximations to the time-dependent transport equation than the other competitive equations. The numerical results confirm that, in the vast majority of cases, the time-dependent SP₂ solutions are significantly more accurate than the time-dependent diffusion and the telegrapher's solutions. We have also shown that the time-dependent SP₂ equations have excellent characteristics such as rotational invariance (which means no ray effect), good diffusion limit behavior, guaranteed positivity in diffusive regimes, and significant accuracy, even in deep-penetration problems. Through computer-running-time tests, we have shown that the time-dependent SP₂ equations can be solved with significantly less computational effort than the conventionally used, time-dependent S_N equations (for N > 2) and almost as fast as the time-dependent diffusion equation. From all these results, we conclude that the time-dependent SP₂ equations should be considered as an important competitor for an improved approximate transport equation solver. Such computationally efficient time-dependent transport models are especially important for problems requiring enhanced computational efficiency, such as neutronics/fluid-dynamics coupled problems that arise in the analyses of hypothetical nuclear reactor accidents.