Iterative solution methods for high-order/hp–DGFEM approximation of the linear Boltzmann transport equation

Houston, Paul; Hubbard, Matthew E.; Radley, Thomas J.

Abstract

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/hp–version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p, we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime.

Citation

Houston, P., Hubbard, M. E., & Radley, T. J. (2024). Iterative solution methods for high-order/hp–DGFEM approximation of the linear Boltzmann transport equation. Computers and Mathematics with Applications, 166, 37-49. https://doi.org/10.1016/j.camwa.2024.04.011

• Journal Article Type: Article
• Acceptance Date: Apr 12, 2024
• Online Publication Date: May 2, 2024
• Publication Date: Jul 15, 2024
• Deposit Date: Apr 15, 2024
• Publicly Available Date: May 3, 2025
• Journal: Computers & Mathematics with Applications
• Print ISSN: 0898-1221
• Electronic ISSN: 1873-7668
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 166
• Pages: 37-49
• DOI: https://doi.org/10.1016/j.camwa.2024.04.011
• Keywords: Linear Boltzmann transport equation; Discontinuous Galerkin methods; Iterative solvers; GMRES; hp–Finite element methods
• Public URL: https://nottingham-repository.worktribe.com/output/33826375
• Additional Information: This article is maintained by: Elsevier; Article Title: Iterative solution methods for high-order/hp–DGFEM approximation of the linear Boltzmann transport equation; Journal Title: Computers & Mathematics with Applications; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.camwa.2024.04.011; Content Type: article; Copyright: © 2024 The Author(s). Published by Elsevier Ltd.