Discontinuous Galerkin Methods for Computational Radiation Transport
Author(s)
Merton, Simon Richard
Type
Thesis
Abstract
This Thesis demonstrates advanced new discretisation technologies that improve
the accuracy and stability of the discontinuous Galerkin finite element method applied
to the Boltzmann Transport Equation, describing the advective transport of
neutral particles such as photons and neutrons within a domain. The discontinuous
Galerkin method in its standard form is susceptible to oscillation detrimental to
the solution. The discretisation schemes presented in this Thesis enhance the basic
form with linear and non-linear Petrov Galerkin methods that remove these oscillations.
The new schemes are complemented by an adjoint-based error recovery
technique that improves the standard solution when applied to goal-based functional
and eigenvalue problems. The chapters in this Thesis have been structured
to be submitted individually for journal publication, and are arranged as follows.
Chapter 1 outlines the Thesis and contains a brief literature review. Chapter 2 introduces
the underlying space-angle discretisation method used in the work, and
discusses a series of potential modifications to the standard discontinuous Galerkin
method. These differ in how the upwinding is performed on the element boundary,
and comprise an upwind-average method, a Petrov-Galerkin method that removes
oscillation by adding artificial diffusion internal to an element and a more sophisticated
Petrov-Galerkin scheme that adds dissipation in the coupling between each
element. These schemes are tested in one-dimension and Taylor analysis of their
convergence rate is included. The chapter concludes with selection of one of the
schemes to be developed in the next part of the Thesis. Chapter 3 develops the
selected method extending it to multi-dimensions. The result is a new discontinuous
Petrov-Galerkin method that is residual based and removes unwanted oscillation
from the transport solution by adding numerical dissipation internal to an
element. The method uses a common length scale in the upwind term for all elements.
This is not always satisfactory, however, as it gives the same magnitude
and type of dissipation everywhere in the domain. The chapter concludes by recommending
some form of non-linearity be included to address this issue. Chapter
4 adds non-linearity to the scheme. This projects the streamline direction, in which
the dissipation acts, onto the solution gradient direction. It defines locally the optimal
amount of dissipation needed in the discretisation. The non-linear scheme is
tested on a variety of steady-state and time-dependent transport problems. Chapters
5, 6 and 7 develop an adjoint-based error measure to complement the scheme
in functional and eigenvalue problems. This is done by deriving an approximation
to the error in the the bulk functional or eigenvalue, and then removing it from the
calculated value in a post-process defect iteration. This is shown to dramatically
accelerate mesh convergence of the goal-based functional or eigenvalue. Chapter
8 concludes the Thesis with recommendations for a further plan of work.
the accuracy and stability of the discontinuous Galerkin finite element method applied
to the Boltzmann Transport Equation, describing the advective transport of
neutral particles such as photons and neutrons within a domain. The discontinuous
Galerkin method in its standard form is susceptible to oscillation detrimental to
the solution. The discretisation schemes presented in this Thesis enhance the basic
form with linear and non-linear Petrov Galerkin methods that remove these oscillations.
The new schemes are complemented by an adjoint-based error recovery
technique that improves the standard solution when applied to goal-based functional
and eigenvalue problems. The chapters in this Thesis have been structured
to be submitted individually for journal publication, and are arranged as follows.
Chapter 1 outlines the Thesis and contains a brief literature review. Chapter 2 introduces
the underlying space-angle discretisation method used in the work, and
discusses a series of potential modifications to the standard discontinuous Galerkin
method. These differ in how the upwinding is performed on the element boundary,
and comprise an upwind-average method, a Petrov-Galerkin method that removes
oscillation by adding artificial diffusion internal to an element and a more sophisticated
Petrov-Galerkin scheme that adds dissipation in the coupling between each
element. These schemes are tested in one-dimension and Taylor analysis of their
convergence rate is included. The chapter concludes with selection of one of the
schemes to be developed in the next part of the Thesis. Chapter 3 develops the
selected method extending it to multi-dimensions. The result is a new discontinuous
Petrov-Galerkin method that is residual based and removes unwanted oscillation
from the transport solution by adding numerical dissipation internal to an
element. The method uses a common length scale in the upwind term for all elements.
This is not always satisfactory, however, as it gives the same magnitude
and type of dissipation everywhere in the domain. The chapter concludes by recommending
some form of non-linearity be included to address this issue. Chapter
4 adds non-linearity to the scheme. This projects the streamline direction, in which
the dissipation acts, onto the solution gradient direction. It defines locally the optimal
amount of dissipation needed in the discretisation. The non-linear scheme is
tested on a variety of steady-state and time-dependent transport problems. Chapters
5, 6 and 7 develop an adjoint-based error measure to complement the scheme
in functional and eigenvalue problems. This is done by deriving an approximation
to the error in the the bulk functional or eigenvalue, and then removing it from the
calculated value in a post-process defect iteration. This is shown to dramatically
accelerate mesh convergence of the goal-based functional or eigenvalue. Chapter
8 concludes the Thesis with recommendations for a further plan of work.
Date Issued
2012-06
Date Awarded
2012-08
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Barrett, David
Smedley-Stevenson, Richard
Pain, Christopher
Publisher Department
Earth Science and Engineering
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)