We present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin (DG) discretizations of the $S_N$ transport equation, and we demonstrate the compatibility of this approach with the Variable Eddington Factor (VEF) method [1], [2]. The new fixup approach is sweep-compatible: during a transport sweep (block Gauss-Seidel iteration in which the scattering source is lagged), a local quadratic programming (QP) problem is solved in each spatial element to ensure that the solution satisfies certain physical constraints, including local particle balance. In this paper, we describe two choices of physical constraints, resulting in two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). In QPZ, the finite element coefficients of the solution are constrained to be nonnegative. In QPMP, they are constrained to adhere to an approximate discrete maximum principle.

There are two primary takeaways in this paper. First, when the positive Bernstein basis is used for DG discretization, the QPMP method eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The latter feature – the dampening of unphysical oscillations – is an improvement upon standard, simpler fixup approaches such as the approach described in [3] (denoted as the “zero and rescale” (ZR) method in this paper). This improvement comes at a moderate computational cost, but it is not prohibitive. Our results show that, even in an unrealistic worst-case scenario where 83% of the spatial elements require a fixup, the computational cost of performing a transport sweep with fixup is only ∼31% greater than performing one without fixup.

The second takeaway is that the VEF method can be used to accelerate the convergence of transport sweeps even when a fixup is applied. When optically thick regions are present, transport sweeps converge slowly, regardless of whether a fixup is applied, and acceleration is needed. However, attempting to apply standard diffusion synthetic acceleration (DSA) to fixed-up transport sweeps results in divergence for optically thick problems. Our results show that the same is not true for VEF. 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.

我们提出了一种新的通量固定方法，用于任意高阶不连续Galerkin（DG）离散化。 ${\mathrm{小号}}_{ñ}$输运方程，我们证明了该方法与可变爱丁顿因子（VEF）方法[1]，[2]的兼容性。新的修正方法与扫描兼容：在传输扫描（块高斯-赛德尔迭代，其中散射源滞后）中，在每个空间元素中解决了局部二次规划（QP）问题，以确保解决方案满足特定的物理条件约束，包括局部粒子平衡。在本文中，我们描述了两种物理约束选择，从而导致了该方法的两种变体：QP零（QPZ）和QP最大原理（QPMP）。在QPZ中，解的有限元系数被约束为非负。在QPMP中，它们必须遵守近似离散的最大原理。

本文主要有两个要点。首先，当将正伯恩斯坦基础用于DG离散化时，QPMP方法消除了负性，为平滑问题保留了高阶精度，并显着抑制了解决方案中的非物理振荡。后者的功能-抑制非物理振荡-是对标准，更简单的固定方法（例如，在[3]中描述的方法（在本文中称为“零和重新缩放”（ZR）方法））的改进。这种改进的代价是适度的计算成本，但这不是禁止的。我们的结果表明，即使在不现实的最坏情况下，其中83％的空间元素都需要固定，执行带固定的运输扫描的计算成本也只比不固定不固定的运输成本高31％。

第二个要点是，即使应用了固定方法，VEF方法也可以用于加速传输扫描的收敛。当存在光学上较厚的区域时，无论是否应用固定，传输扫描都会缓慢收敛，并且需要加速。但是，尝试将标准扩散合成加速度（DSA）应用于固定的传输扫描会导致光学厚度问题产生发散。我们的结果表明，VEF并非如此。当将VEF与固定运输扫描组合使用时，结果是产生非负解的方案，独立于平均自由程收敛，并且在QPMP固定的情况下，遵循近似离散的最大原理。