Accurate solutions to time dependent transport problems with a moving mesh and exact uncollided source treatment

TL;DR

This paper introduces a novel Discontinuous Galerkin method with a moving mesh and exact uncollided source treatment for time-dependent transport problems, achieving high accuracy and spectral convergence especially in smooth scenarios.

Contribution

The paper presents a new DG-based numerical method with a moving mesh and uncollided source treatment that improves accuracy and convergence in time-dependent transport simulations.

Findings

Achieves spectral convergence on smooth problems.

Significantly reduces error in nonsmooth source problems.

Outperforms standard DG in accuracy, especially for smooth solutions.

Abstract

For the purpose of finding benchmark quality solutions to time dependent Sn transport problems, we develop a numerical method in a Discontinuous Galerkin (DG) framework that utilizes time dependent cell edges, which we call a moving mesh, and an uncollided source treatment. The DG method for discretizing space is a powerful solution technique on smooth problems and is robust on non-smooth problems. In order to realize the potential of the DG method to spectrally resolve smooth problems, our moving mesh and uncollided source treatment is devised to circumvent discontinuities in the solution or the first derivative of the solutions that are admitted in transport calculations. The resulting method achieves spectral convergence on smooth problems, like a standard DG implementation. When applied to problems with nonsmooth sources that induce discontinuities, our moving mesh, uncollided source method returns a significantly more accurate solution than the standard DG method. On problems with smooth sources, we observe spectral convergence even in problems with wave fronts. In problems where the angular flux is inherently non-smooth, as in Ganapol's (2001) well known plane pulse benchmark, we do not observe an elevated order of accuracy when compared with static meshes, but there is a reduction in error that is nearly three orders of magnitude.