Abstract

The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error.

In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. A metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity.

Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.

摘要

离散纵坐标线性玻尔兹曼输运方程通常以其空间离散形式求解，导致空间离散误差。出于自适应网格细化或误差分析等目的而对该误差进行量化需要后验估计器，该估计器利用空间离散方程的数值解来计算估计值。因为数值解的质量决定了误差估计，所以任何实际问题配置的真实解中存在的不规则性往往会导致误差估计与真实误差的最大偏差。

在本文中，用于减少估计器误差的分析部分奇异特征跟踪 (pSCT) 程序在我们的零阶不连续 Galerkin 方案的新型残差源估计器中实现，但增加了单次内部迭代的成本。pSCT 方案与标准剩余源估计器的基于度量的评估是在制造解决方案测试套件的参数范围内执行的。pSCT 方案在估计器误差的主要来源是解不规则性的问题（即真解不连续的问题和真解的一阶导数不连续且散射比为低的。

最终，在计算成本合理的问题中，pSCT 方案被认为可用于误差估计。在存在许多不规则性的情况下，这种方案可能难以普遍使用，但在基准测试中，作为一种分析工具，或在具有非耗散不连续性的问题中，该方案可能证明是无价的。