Solution Irregularity Remediation for Spatial Discretization Error Estimation for S_N Transport Solutions

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.

Keywords:

• Spatial discretization
• error estimation
• solution regularity
• singular characteristics

Acknowledgments

This work is supported by the U.S. Department of Energy National Nuclear Security Administration under award number DENA0002576. We wish to thank Jose Duo for his consultation on SCT.

Notes

^a In realistic problems, $r$ is at most $\frac{3}{2}-\xi$, where $\xi$ is a very small number.⁴ In the case where the solution is discontinuous, $r$ is $\frac{1}{2}-\xi$.

^b North and south boundaries are vacuum. West and east boundaries have ${\mathrm{\Psi }}_m(x,y)=9$.

^c $\mathrm{\Delta }$ = generic cell width assuming uniform refinement across spatial dimensions.