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Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

Paul Houston, Matthew E. Hubbard, Thomas J. Radley, Oliver J. Sutton, Richard S. J. Widdowson

TL;DR

This work addresses the high dimensional challenge of the stationary linear Boltzmann transport equation by introducing an hp-version discontinuous Galerkin finite element method that discretises space, angle, and energy on a unified framework using polytopic meshes. The authors derive hp-based approximation and stability results, including an inf-sup bound, and show that the scheme can be efficiently implemented as a multigroup discrete ordinates method by exploiting energy and angular decoupling through carefully chosen bases and quadrature. The key contributions are the hp a priori error analysis, the novel basis construction that enables fast energy and angle integration, and a practical algorithm that achieves high order without sacrificing parallel performance. The approach is well suited for complex geometries and enables arbitrary order accuracy in Boltzmann transport simulations, with numerical results in polyenergetic 2D and monoenergetic 3D settings validating the theory and demonstrating potential practical impact in medical physics and reactor design.

Abstract

We introduce an $hp$-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and $hp$-version a priori error analysis of the proposed method, by deriving suitable $hp$-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

TL;DR

This work addresses the high dimensional challenge of the stationary linear Boltzmann transport equation by introducing an hp-version discontinuous Galerkin finite element method that discretises space, angle, and energy on a unified framework using polytopic meshes. The authors derive hp-based approximation and stability results, including an inf-sup bound, and show that the scheme can be efficiently implemented as a multigroup discrete ordinates method by exploiting energy and angular decoupling through carefully chosen bases and quadrature. The key contributions are the hp a priori error analysis, the novel basis construction that enables fast energy and angle integration, and a practical algorithm that achieves high order without sacrificing parallel performance. The approach is well suited for complex geometries and enables arbitrary order accuracy in Boltzmann transport simulations, with numerical results in polyenergetic 2D and monoenergetic 3D settings validating the theory and demonstrating potential practical impact in medical physics and reactor design.

Abstract

We introduce an -version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and -version a priori error analysis of the proposed method, by deriving suitable -approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.
Paper Structure (18 sections, 12 theorems, 89 equations, 2 figures, 1 algorithm)

This paper contains 18 sections, 12 theorems, 89 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Model problem
  4. Discrete scheme
  5. Spatial discretisation
  6. Angular discretisation
  7. Energy discretisation
  8. Discontinuous Galerkin scheme
  9. Inverse inequalities and approximation theory
  10. Stability and convergence of the discrete scheme
  11. Efficient implementation as a multigroup discrete ordinates scheme
  12. Multigroup implementation in energy
  13. Discrete ordinates implementation in angle
  14. Full algorithm
  15. Numerical results
  16. ...and 3 more sections

Key Result

Lemma 6

Let $\kappa_{\tiny\Omega}\in\mathcal{T}_{\Omega}$, $F\subset \partial\kappa_{\tiny\Omega}$ denote one of its faces. Then, for each $v\in\mathbb{P}_p(\kappa_{\tiny\Omega})$, we have the inverse estimate where $C_{\rm inv}^{F}$ is a positive constant, which depends on the shape regularity of the covering of $\kappa_{\tiny\Omega}$, if $\kappa_{\tiny\Omega} \in \tilde{\mathcal{T}_{\Omega}}$, but is i

Figures (2)

  • Figure 1: Example 1: Convergence of the DGFEM under $h$--refinement for $p=0,1,2$. Here, the DGFEM-norm is defined in \ref{['eq:dGnorm']}.
  • Figure 2: Example 2: Convergence of the method under $h$--refinement for $p=0,1,2$. Here, the DGFEM-norm is defined in \ref{['eq:dGnorm']}.

Theorems & Definitions (21)

  • Remark 1
  • Definition 4
  • Definition 5
  • Lemma 6
  • Lemma 8: CaDoGeHo2017 (Lemma 14)
  • Lemma 9
  • Definition 10
  • Theorem 11
  • Lemma 12
  • Lemma 13
  • ...and 11 more