  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.
