Diffusion Accelerated Implicit Monte Carlo via Nonlinear Elimination for Thermal Radiative Transfer

In this dissertation, we derive and implement a new transport-diffusion hybrid algorithm for solving thermal radiative transfer (TRT) problems. Using the method of nonlinear elimination (NLEM), the TRT system of equations can be written in terms of a transport equation with the absence of scattering and a diffusion equation. The transport solution is obtained using a Monte Carlo (MC) method with implicit capture and the diffusion solution is used to accelerate the transport convergence. We name this method Diffusion Accelerated Implicit Monte Carlo (DAIMC). A series of tests are used to verify the proposed algorithm and its associated solvers. After the verification of DAIMC, we investigate its performance by comparing DAIMC results to those obtained from the traditional Implicit Monte Carlo (IMC) method. In 1D slab geometry calculations, we show that DAIMC yields a more accurate solution than IMC when compared to the analytic solution. The increased accuracy of the DAIMC solution comes at the cost of an increased computational time when compared to IMC. We have also employed Quasi-Monte Carlo (QMC) in the DAIMC algorithm for 1D calculations. QMC retains the same accuracy as the MC implementation of DAIMC while decreasing the required computing time. We also implemented DAIMC in 2D-XY geometry using a piecewise constant representation of temperatures for the Monte Carlo transport solver and a linear-continuous discretization for the diffusion equation. For problems in which the opacity is constant or has a T^{-1} temperature dependence, the implementation choice for DAIMC converges to the correct equilibrium solution and provides more accurate results than the IMC method. We observed that small time steps are required for DAIMC to produce the analytic equilibrium solution when the opacity has a temperature dependence of T^{-2}. DAIMC results for a crooked pipe problem are compared with results obtained from the IMC method. We observed nonphysical overheating at the interface of the thick and thin material region for both our DAIMC method and the IMC method. The nonphysical overheating of the interface improves with refinement of the mesh for both methods

Higher Order Discontinuous Finite Element Methods for Discrete Ordinates Thermal Radiative Transfer

The linear discontinuous finite element method (LDFEM) is the current work horse of the radiation transport community. The popularity of LDFEM is a result of LDFEM (and its Q1 multi-dimensional extensions) being both accurate and preserving the thick diffusion limit. In practice, the LDFEM equations must be “lumped” to mitigate negative radiation transport solutions. Negative solutions are non-physical, but are inherent to the mathematics of LDFEM and other spatial discretizations. Ongoing changes in high performance computing (HPC) are dictating a preference for increased numbers of floating point operations (FLOPS) per unknown. Higher order discontinuous finite element methods (DFEM), those with polynomial trial spaces greater than linear, have been found to offer more accuracy per unknown than LDFEM. However, DFEM with higher degree trial spaces have received only limited attention due to their increased computational time per unknown, LDFEM's preservation of the thick diffusion limit, and the relative accuracy of LDFEM compared to other historical spatial discretizations. As solution methods evolve to make the most efficient use of HPC, it is possible that the increased computational work of higher order DFEM may become a strength rather than a hindrance. For higher order DFEM to be useful in practice, lumping techniques must be developed to inhibit negative radiation transport solutions. We will show that traditional mass matrix lumping does not guarantee positive solutions and limits the overall accuracy of the DFEM scheme. To solve this problem, we propose a new, quadrature based, self-lumping technique. Our self-lumping technique does not limit solution order of convergence, improves solution positivity, and can be easily adapted to account for the within cell variation of interaction cross section. To test and demonstrate the characteristics of our self-lumping methodology, we apply our schemes to several test problems: a homogeneous, source-free pure absorber; a pure absorber with spatially varying cross section; a model fuel depletion problem; and finally, we solve the grey thermal radiative transfer equations

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described