Wang, Yaqi (2009-05). Adaptive Mesh Refinement Solution Techniques for the Multigroup SN Transport Equation Using a Higher-Order Discontinuous Finite Element Method. Doctoral Dissertation.
In this dissertation, we develop Adaptive Mesh Refinement (AMR) techniques
for the steady-state multigroup SN neutron transport equation using a higher-order
Discontinuous Galerkin Finite Element Method (DGFEM). We propose two error estimations,
a projection-based estimator and a jump-based indicator, both of which
are shown to reliably drive the spatial discretization error down using h-type AMR.
Algorithms to treat the mesh irregularity resulting from the local refinement are
implemented in a matrix-free fashion. The DGFEM spatial discretization scheme
employed in this research allows the easy use of adapted meshes and can, therefore,
follow the physics tightly by generating group-dependent adapted meshes. Indeed,
the spatial discretization error is controlled with AMR for the entire multigroup SNtransport
simulation, resulting in group-dependent AMR meshes. The computing
efforts, both in memory and CPU-time, are significantly reduced. While the convergence
rates obtained using uniform mesh refinement are limited by the singularity
index of transport solution (3/2 when the solution is continuous, 1/2 when it is discontinuous),
the convergence rates achieved with mesh adaptivity are superior. The
accuracy in the AMR solution reaches a level where the solution angular error (or ray
effects) are highlighted by the mesh adaptivity process. The superiority of higherorder
calculations based on a matrix-free scheme is verified on modern computing architectures.
A stable symmetric positive definite Diffusion Synthetic Acceleration (DSA)
scheme is devised for the DGFEM-discretized transport equation using a variational
argument. The Modified Interior Penalty (MIP) diffusion form used to accelerate the
SN transport solves has been obtained directly from the DGFEM variational form of
the SN equations. This MIP form is stable and compatible with AMR meshes. Because
this MIP form is based on a DGFEM formulation as well, it avoids the costly
continuity requirements of continuous finite elements. It has been used as a preconditioner
for both the standard source iteration and the GMRes solution technique
employed when solving the transport equation. The variational argument used in
devising transport acceleration schemes is a powerful tool for obtaining transportconforming
xuthus, a 2-D AMR transport code implementing these findings, has been developed
for unstructured triangular meshes.