Discontinuous Galerkin Methods for Computational Radiation Transport

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.