Asymptotic derivation of the time-dependent SP2 equations and numerical calculations

Converting the independent variables of the transport equation to dimensionless parameters, asymptotic analyses can be performed to show that for an important class of problems, the diffusion equation is an asymptotic limit of the transport equation. A recent paper by Larsen et al. provides a broadened view of the aforementioned result. It deals with the steady state transport equation and shows that the simplified spherical-harmonics (SPN) equations are robust high-order asymptotic approximations of the transport equation in a physical regime in which the conventional diffusion equation is the leading-order approximation. According to the reported numerical results for the steady-state cases for many problems, low-order SPN equations capture most (Gamino reports open-quotes greater than 80%close quotes) of the transport corrections to the diffusion approximation. Tomasevic and Larsen show that in nearly all cases, the SP2 results are significantly more accurate than diffusion results in the steady-state problems. In this paper, we deal with the time-dependent transport equation using conventional asymptotic approaches and find that the time-dependent SP2 equations have the same asymptotic approximations as the time-dependent transport equation up to the third order in a physical regime in which the time-dependent diffusion equation is the leading-order approximation. In addition, we follow the same asymptotic approaches as Larsen et al. and find that if one neglects the time derivative term of the second moment of angular flux ∂ψ2/∂t in the time-dependent SP2 equations, an equation is obtained that is identical to the third-order asymptotic approximation of the time-dependent transport equation (we will call this equation the modified time-dependent SP2 equation)