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

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.Comment: 27 pages, 2 figure

Analysis of Iterative Methods for the Linear Boltzmann Transport Equation

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 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 discretisation parameters. 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.Comment: 27 pages, 8 figure

Adaptive mesh refinement techniques for diffusion-synthetic-accelerated discrete-ordinates neutral particle transport

An Adaptive Mesh Refinement (AMR) technique is presented for the one-group and the multigroup SN transport equations discretized using a Discontinuous Galerkin (DG) method. A diffusion synthetic accelerator, also based on a DG discretization and directly obtained from the discretized transport equations, is given. Numerical results are provided for 2D unstructured triangular meshes

Discontinuous Galerkin FEMs for Radiation Transport Problems

The linear Boltzmann transport equation (LBTE), a high dimensional partial integrodifferential equation, is used to model radiation transport. Radiation transport is an area of physics that is concerned with the propagation and distribution of radiative particles, such as photons and electrons within a material medium. In this thesis, we present a high-order discontinuous Galerkin finite element method (DGFEM) discretisations of the steady state linear Boltzmann transport equation in the spatial, angular, and energy domains. Comparisons between the tensor discontinuous Galerkin finite element method and discrete ordinates method show that the former is higher order than the latter. A method of block diagonalising the resulting matrix into a sequence of transport equations coupled by the right hand side while retaining high order convergence, is demonstrated for both the angular and energy domains. This new method offers the arbitrary order convergence rates of the discontinuous Galerkin finite element method, but it can be implemented in an almost identical form to standard multigroup discrete ordinates methods. The assembly of the matrix for the resulting transport equations for a variety of different type of elements is discussed. The generation of meshes formed of general polytopes is discussed, and a comparison between the time to solve the transport equation on meshes formed of the different element types follows. An efficient implementation of the discontinuous Galerkin finite element method for transport equations is then presented. This algorithm exploits the fixed wind direction of the transport problems resulting from the discontinuous Galerkin finite element method of the LBTE, to solve the transport problem while never forming the matrix. We then compare this algorithm to a direct matrix solver for both convex and non-convex polytopes. An a posteriori error bound for the discontinuous Galerkin finite element method of the LBTE and the transport problem is then derived. We use this error bound to develop an adaptive framework for the LBTE. Three different adaptive algorithms for the LBTE are then presented and compared. The h-refinement algorithm, which marks the element with the error of each tensor element it is part of, shows a clear advantage over the other methods

Discontinuous Galerkin FEMs for Radiation Transport Problems

The linear Boltzmann transport equation (LBTE), a high dimensional partial integrodifferential equation, is used to model radiation transport. Radiation transport is an area of physics that is concerned with the propagation and distribution of radiative particles, such as photons and electrons within a material medium. In this thesis, we present a high-order discontinuous Galerkin finite element method (DGFEM) discretisations of the steady state linear Boltzmann transport equation in the spatial, angular, and energy domains. Comparisons between the tensor discontinuous Galerkin finite element method and discrete ordinates method show that the former is higher order than the latter. A method of block diagonalising the resulting matrix into a sequence of transport equations coupled by the right hand side while retaining high order convergence, is demonstrated for both the angular and energy domains. This new method offers the arbitrary order convergence rates of the discontinuous Galerkin finite element method, but it can be implemented in an almost identical form to standard multigroup discrete ordinates methods. The assembly of the matrix for the resulting transport equations for a variety of different type of elements is discussed. The generation of meshes formed of general polytopes is discussed, and a comparison between the time to solve the transport equation on meshes formed of the different element types follows. An efficient implementation of the discontinuous Galerkin finite element method for transport equations is then presented. This algorithm exploits the fixed wind direction of the transport problems resulting from the discontinuous Galerkin finite element method of the LBTE, to solve the transport problem while never forming the matrix. We then compare this algorithm to a direct matrix solver for both convex and non-convex polytopes. An a posteriori error bound for the discontinuous Galerkin finite element method of the LBTE and the transport problem is then derived. We use this error bound to develop an adaptive framework for the LBTE. Three different adaptive algorithms for the LBTE are then presented and compared. The h-refinement algorithm, which marks the element with the error of each tensor element it is part of, shows a clear advantage over the other methods

Discontinuous Galerkin Methods for the Linear Boltzmann Transport Equation

Radiation transport is an area of applied physics that is concerned with the propagation and distribution of radiative particle species such as photons and electrons within a material medium. Deterministic models of radiation transport are used in a wide range of problems including radiotherapy treatment planning, nuclear reactor design and astrophysics. The central object in many such models is the (linear) Boltzmann transport equation, a high-dimensional partial integro-differential equation describing the absorption, scattering and emission of radiation. In this thesis, we present high-order discontinuous Galerkin finite element discretisations of the time-independent linear Boltzmann transport equation in the spatial, angular and energetic domains. Efficient implementations of the angular and energetic components of the scheme are derived, and the resulting method is shown to converge with optimal convergence rates through a number of numerical examples. The assembly of the spatial scheme on general polytopic meshes is discussed in more detail, and an assembly algorithm based on employing quadrature-free integration is introduced. The quadrature-free assembly algorithm is benchmarked against a standard quadrature-based approach, and an analysis of the algorithm applied to a more general class of discontinuous Galerkin discretisations is performed. In view of developing efficient linear solvers for the system of equations resulting from our discontinuous Galerkin discretisation, we exploit the variational structure of the scheme to prove convergence results and derive a posteriori solver error estimates for a family of iterative solvers. These a posteriori solver error estimators can be used alongside standard implementations of the generalised minimal residual method to guarantee that the linear solver error between the exact and approximate finite element solutions (measured in a problem-specific norm) is below a user-specified tolerance. We discuss a family of transport-based preconditioners, and our linear solver convergence results are benchmarked through a family of numerical examples

Higher-Order DGFEM Transport Calculations on Polytope Meshes for Massively-Parallel Architectures

In this dissertation, we develop improvements to the discrete ordinates (S_N) neutron transport equation using a Discontinuous Galerkin Finite Element Method (DGFEM) spatial discretization on arbitrary polytope (polygonal and polyhedral) grids compatible for massively-parallel computer architectures. Polytope meshes are attractive for multiple reasons, including their use in other physics communities and their ease in handling local mesh refinement strategies. In this work, we focus on two topical areas of research. First, we discuss higher-order basis functions compatible to solve the DGFEM S_N transport equation on arbitrary polygonal meshes. Second, we assess Diffusion Synthetic Acceleration (DSA) schemes compatible with polytope grids for massively-parallel transport problems. We first utilize basis functions compatible with arbitrary polygonal grids for the DGFEM transport equation. We analyze four different basis functions that have linear completeness on polygons: the Wachspress rational functions, the PWL functions, the mean value coordinates, and the maximum entropy coordinates. We then describe the procedure to extend these polygonal linear basis functions into the quadratic serendipity space of functions. These quadratic basis functions can exactly interpolate monomial functions up to order 2. Both the linear and quadratic sets of basis functions preserve transport solutions in the thick diffusion limit. Maximum convergence rates of 2 and 3 are observed for regular transport solutions for the linear and quadratic basis functions, respectively. For problems that are limited by the regularity of the transport solution, convergence rates of 3/2 (when the solution is continuous) and 1/2 (when the solution is discontinuous) are observed. Spatial Adaptive Mesh Refinement (AMR) achieved superior convergence rates than uniform refinement, even for problems bounded by the solution regularity. We demonstrated accuracy in the AMR solutions by allowing them to reach a level where the ray effects of the angular discretization are realized. Next, we analyzed DSA schemes to accelerate both the within-group iterations as well as the thermal upscattering iterations for multigroup transport problems. Accelerating the thermal upscattering iterations is important for materials (e.g., graphite) with significant thermal energy scattering and minimal absorption. All of the acceleration schemes analyzed use a DGFEM discretization of the diffusion equation that is compatible with arbitrary polytope meshes: the Modified Interior Penalty Method (MIP). MIP uses the same DGFEM discretization as the transport equation. The MIP form is Symmetric Positive De_nite (SPD) and e_ciently solved with Preconditioned Conjugate Gradient (PCG) with Algebraic MultiGrid (AMG) preconditioning. The analysis from previous work was extended to show MIP's stability and robustness for accelerating 3D transport problems. MIP DSA preconditioning was implemented in the Parallel Deterministic Transport (PDT) code at Texas A&M University and linked with the HYPRE suite of linear solvers. Good scalability was numerically verified out to around 131K processors. The fraction of time spent performing DSA operations was small for problems with sufficient work performed in the transport sweep (O(10^3) angular directions). Finally, we have developed a novel methodology to accelerate transport problems dominated by thermal neutron upscattering. Compared to historical upscatter acceleration methods, our method is parallelizable and amenable to massively parallel transport calculations. Speedup factors of about 3-4 were observed with our new method

Discontinuous Galerkin Methods for the Linear Boltzmann Transport Equation

Radiation transport is an area of applied physics that is concerned with the propagation and distribution of radiative particle species such as photons and electrons within a material medium. Deterministic models of radiation transport are used in a wide range of problems including radiotherapy treatment planning, nuclear reactor design and astrophysics. The central object in many such models is the (linear) Boltzmann transport equation, a high-dimensional partial integro-differential equation describing the absorption, scattering and emission of radiation. In this thesis, we present high-order discontinuous Galerkin finite element discretisations of the time-independent linear Boltzmann transport equation in the spatial, angular and energetic domains. Efficient implementations of the angular and energetic components of the scheme are derived, and the resulting method is shown to converge with optimal convergence rates through a number of numerical examples. The assembly of the spatial scheme on general polytopic meshes is discussed in more detail, and an assembly algorithm based on employing quadrature-free integration is introduced. The quadrature-free assembly algorithm is benchmarked against a standard quadrature-based approach, and an analysis of the algorithm applied to a more general class of discontinuous Galerkin discretisations is performed. In view of developing efficient linear solvers for the system of equations resulting from our discontinuous Galerkin discretisation, we exploit the variational structure of the scheme to prove convergence results and derive a posteriori solver error estimates for a family of iterative solvers. These a posteriori solver error estimators can be used alongside standard implementations of the generalised minimal residual method to guarantee that the linear solver error between the exact and approximate finite element solutions (measured in a problem-specific norm) is below a user-specified tolerance. We discuss a family of transport-based preconditioners, and our linear solver convergence results are benchmarked through a family of numerical examples