Multi-dimensional High-order Discretizations and Fast Solution Algorithms for Simplified Spherical Harmonic Approximations
Kabelitz, Matthew
2023
Abstract
Efficient and flexible algorithms for solving low-order transport problems are highly desirable in a great number of applications. Such solvers are useful not only as rapid design and evaluation tools in their own right, but as acceleration techniques to be used in larger, more precise simulations. The focus of this thesis is a novel discretization and solver for the elliptic simplifications to the neutron transport equation; encompassing diffusion, Simplified $P_N$ ($SP_N$) methods, and Generalized Simplified $P_N$ which is the evolution of new theory pertaining to the $SP_N$ boundary conditions. Demonstrations on a 1D test problem show that our new 4th-order Legendre-Gauss-Lobatto discretization has the same $h$-convergance (5th order) as traditional NEM, achieving the maximum convergence order imposed by the Bramble-Hilbert lemma. Unlike more traditional finite element methods, our discretization also preserves continuity at interfaces, and minimizes the number of basis functions used to represent these conditions. Our numerical results also show that this order of convergence extends to multidimensional problems without the need for transverse integration, and is capable of being applied to certain kinds of unstructured mesh. The system of equations this discretization produces are also amenable to simplification and solution via hierarchical fast solvers such as the Hierarchical Poincaré-Steklov Method. This method contains explicit representations of certain conserved quantities, making it attractive for an extension of the 1D nodal method into multiple dimensions. This hierarchical method is implemented in a proof-of-concept multigroup code with the capability to handle deformed structured meshes, with no spatial homogenization. The solver utilizes no transverse leakage approximation in its multidimensional calculations, and is designed to operate on a sub-pin mesh. An eigenvalue solver based on power iteration is also implemented. Clear avenues exist for the acceleration of this solver with structured linear algebra packages, as well as the expansion from the deformed grid to a truly unstructured mesh. This solver is verified using a Method of Manufactured Solutions order-of-accuracy study which proves that maximal convergence order is still obtainable. Additionally, eigenvalue results are presented for 2D problems derived from the common benchmark C5G7, and are compared with reference obtained from the Method of Characteristics code MPACT. This analysis reveals typical errors characteristic of all methods based upon a spherical harmonics representation of the angular flux, further indicating proper functioning of the solver procedure. Overall, we establish a new discretization and solver framework for low-order transport equations. Future work may focus on developing high performance parallel implementations, more numerically robust implementations, and more generic unstructured mesh implementations.Deep Blue DOI
Subjects
Fast Direct Solver Simplified PN Methods Nodal Methods
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