A Quadratic Programming Flux Correction Method for High-Order DG Discretizations of SN Transport

TL;DR

This paper introduces a quadratic programming flux correction method for high-order DG discretizations of the SN transport equation, ensuring physical constraints, high accuracy, and improved convergence in optically thick regions.

Contribution

It proposes a novel QP-based flux fixup approach compatible with high-order DG methods, including two variants that enforce physical constraints and improve solution quality.

Findings

QPMP eliminates negativities and reduces oscillations.

The method preserves high-order accuracy for smooth problems.

Combining VEF with fixup accelerates convergence in thick regions.

Abstract

We present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin discretizations of the SN transport equation. This approach is sweep-compatible: as the transport sweep is performed, a local quadratic programming (QP) problem is solved in each spatial cell to ensure that the solution satisfies certain physical constraints, including local particle balance. The constraints can be chosen in two ways, leading to two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). The coefficients of the solution are constrained to be nonnegative in QPZ, and they are constrained by an approximate discrete maximum principle in QPMP. There are two primary takeaways in this paper. First, it is shown that the QPMP method, when used with the positive Bernstein basis, eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The second takeaway is that the Variable Eddington Factor (VEF) method can be used to accelerate the convergence of source iteration with fixup for problems with optically thick regions. Regardless of whether a fixup is applied, source iteration converges slowly when optically thick regions are present and acceleration is needed. When VEF is combined with fixed-up transport sweeps, the result is a scheme that produces a nonnegative solution, converges independently of the mean free path, and, in the case of the QPMP fixup, adheres to an approximate discrete maximum principle.