A Linear, Exponential-Discontinuous Scheme for Discrete-Ordinates Calculations in Slab Geometry

TL;DR

The paper addresses accurate deterministic SN solutions in slab geometry while preserving positivity and the diffusion-limit by introducing a strictly linear, discontinuous-Petrov-Galerkin discretization. It employs a constant+exponential trial basis with a constant+linear test basis to form the EX scheme, which is shown to yield improved local errors over linear-discontinuous methods for thin cells and to approach the linear-characteristic solution when that solution is positive, with positivity guaranteed under $|s_1|<3s_0$. Numerical experiments in a deep-penetration slab demonstrate that EX often provides the best scalar flux accuracy among compared schemes, though non-converged results and boundary-based negatives highlight the need for further basis refinements or nonlinear fixes for strong attenuation. The work suggests that the EX approach can extend to higher dimensions and motivate continued development of mixed-basis DG/DPG strategies for neutron transport problems.

Abstract

Presented here is a preliminary study of a strictly linear, discontinuous-Petrov-Galerkin scheme for the discrete-ordinates method in slab geometry. By ``linear'', we mean the discretization does not depend on the solution itself as is the case in classical ``fix-up'' schemes and other nonlinear schemes that have been explored to maintain positive solutions with improved accuracy. By discontinuous, we mean the angular flux $ψ$ and scalar flux $φ$ are piecewise continuous functions that may exhibit discontinuities at cell boundaries. Finally, by ``Petrov-Galerkin,'' we mean a finite-element scheme in which the ``trial'' and ``test'' functions differ. In particular, we find that a trial basis consisting of a constant and exponential function that exactly represents the step-characteristic solution with a constant and linear test basis produces a scheme (1) with slightly better local errors than the linear-discontinuous (LD) scheme (for thin cells), (2) accuracy that approaches the linear-characteristic (LC) scheme (when the LC solution is positive), and (3) is positive as long as the first two source Legendre moments satisfy $|s_1| < 3 s_0$.

Figures (2)

• Figure 1: Results for $\Delta = 5$ cm ($\tau = 10$ between 4 and 6 cm). White markers indicate a negative flux value.
• Figure 2: Results for $\Delta = 2.5$ cm ($\tau = 5$ between 4 and 6 cm). White markers indicate a negative flux value.