High Order Implicit Residual-Based Spatial Discretization Error Estimation for S N Neutron Transport

Abstract

This work demonstrates our novel residual source spatial discretization error estimator (LeR/TEAD) for a DGFEM-1 discretization and assesses it along with two contemporary estimators, Ragusa and Wang's h -refinement estimator (RW) and Duo, Azmy, and Zikatanov's explicit residual-based estimator (DAZ), on a suite of Method of Manufactured Solutions (MMS) 2D problems and three realistic problem geometries. LeR/TE-AD is attractive because it directly estimates the local error in the angular flux, as opposed to a mere indicator of the error's behavior, on the same mesh and method order as the original numerical solution, thus typically being less computationally intensive than a refinement-based method. On the MMS suite, LeR/TE-AD consistently displayed a reduced performance versus its DGFEM-0 results in terms of accuracy and precision metrics, though it was not typically grossly inaccurate. This is attributed to the irregularities in the true solution across singular characteristics limiting the local accuracy of the numerical flux solution, leading to poor derivative approximations used in the residual approximations. The error transport problem then spreads the error in the residual to nearby cells, causing a greater degree of imprecision that did not afflict DAZ or RW. In testing the estimators on realistic problem geometries, however, LeR/TE-AD fared better. In practice, the true error is much larger in non-idealized geometries like in MMS, and a superlinear true solution means that RW and DAZ are not beneficially biased for DGFEM-1 error estimation. LeR/TE-AD was typically first or second in accuracy, primarily competing with RW, but the latter usually consumed 2-4 times the computational time as LeR/TE-AD, and requires a solution with four times as many unknowns. Furthermore, RW and LeR/TE-AD can be used to compute direct estimates of the error in any quantity of interest that is based on the angular ux solution, such as the fission rate density in a fuel pin, whereas DAZ requires a heuristic extension due to its norm-based nature.