Development of High-Order P\u3cem\u3e\u3csub\u3eN\u3c/sub\u3e\u3c/em\u3e Models for Radiative Heat Transfer in Special Geometries and Boundary Conditions

The high-order spherical harmonics () method for 2-D Cartesian domains is extracted from the 3-D formulation. The number of equations and intensity coefficients reduces to (N+1)2/4 in the 2-D Cartesian formulation compared with N(N+1)/2 for the general 3-D formulation. The Marshak boundary conditions are extended to solve problems with nonblack and mixed diffuse-specular surfaces. Additional boundary conditions for specified radiative wall flux, for symmetry/specular reflection boundaries have also been developed. The mathematical details of the formulations and their implementation in the OpenFOAM finite volume based CFD software platform are presented. The accuracy and computational cost of the 2-D Cartesian are compared with that of the 3-D solver and a Photon Monte Carlo solver for a square enclosure, as well as a 45° wedge geometry with variable radiative properties. The new boundary conditions have been applied for both test cases, and the boundary condition for mixed diffuse-specular surfaces is further illustrated by numerical examples of a rectangular geometry enclosed by walls with different surface characteristics

A convergent method for linear half-space kinetic equations

We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main strategy in both analysis and numerics includes three steps: adding damping terms to the original half-space equation, using an inf-sup argument and even-odd decomposition to establish the well-posedness of the damped equation, and then recovering solutions to the original half-space equation. The proposed numerical methods for the damped equation is shown to be quasi-optimal and the numerical error of approximations to the original equation is controlled by that of the damped equation. This efficient solution to the half-space problem is useful for kinetic-fluid coupling simulations

StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent $P_N$ and $SP_N$ equations, of radiative transfer. The method, which works for arbitrary moment order $N$, makes use of the specific coupling between the moments in the $P_N$ equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.Comment: 28 pages, 7 figures; StaRMAP code available at http://www.math.temple.edu/~seibold/research/starma

Convergence acceleration aspects in the solution of the PN neutron transport eigenvalue problem

The solution of the eigenvalue problem for neutron transport is of utmost importance in the field of reactor physics, and represents a challenging problem for numerical models. Different eigenvalue formulations can be identified, each with its own physical significance. The numerical solution of these problems by deterministic methods requires the introduction of approximations, such as the spherical harmonics expansion in PN models, leading to results that depend on the approximations introduced (spatial mesh size, N order, ...). All these results represent, in principle, sequences that can easily profit from acceleration techniques to approach convergence towards the correct value. Such a reference value is estimated, in this work, by the Monte Carlo technique. The Wynn- acceleration method is applied to the various sequences of eigenvalues emerging when tackling the solution of the PN models with different orders and increasing spatial accuracy, in order to obtain more accurate, benchmark-quality results. It is shown that the acceleration can be successfully applied and that the analysis of the results of different acceleration approaches sheds some light on the physical meaning of the numerical approximations