Asymptotic derivation and numerical investigation of time-dependent simplified equations

Abstract

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.

StaRMAP---A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer

“…Moreover, the matrix C(x, y, t) is diagonal, and the matrices M x and M y are constant-coefficient, and possess patterns of their nonzero entries, as described in Lemma 2. Hence, the real-valued P N equations (7) are covered, as are the SP N equations [21]. 4.1.…”

“…Each time step consists of four half steps. The even solution operator (20) updates the components on the even grids G 11 and G 22 , by using spatial derivatives of quantities on the odd grids G 21 and G 21 ; and vice versa for the odd solution operator (21). One step with the even (odd) solution operator consists therefore of two parts: first, spatial differences of the odd (even) components are computed (see above); second, the even (odd) components are updated, via the code segment The update rule for the j th moment is an immediate implementation of the update formula (19).…”

Validation of the SHNC time-dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies

“…In the following, we will only consider L to be an odd integer because, as a consequence of the interface conditions given by Eqs. (13), the even order moments are continuous functions and, in particular, the neutronic scalar flux Φ d Ω = √ 4π φ 00 is continuous along the reactor volume. Defining the real moments…”

“…Later, it was showed [11] that the SP L equations can be understood as an asymptotic approximation of the P L equations. These equations have been successfully applied to other fields [12] and extended to time-dependent problems [13].…”

Massively Parallel Stencil Strategies for Radiation Transport Moment Model Simulations

“…2.1), the StaRMAP approach is implemented as an efficiently vectorized open-source MATLAB code [36]. The software's straightforward usability and flexibility have made it a popular research tool, used in particular in numerous dissertations, and it has been extended to other moment models (filtered [16] and simplified [33]), and applied in radiotherapy simulations [12,19]. For 2D problems, the vectorized MATLAB implementation allows for serial or shared memory (MATLAB's automatic usage of multiple cores) parallel execution speeds that are on par with comparable implementations of the methodology in C++.…”