SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page B1271-B1301, January 2020.
This paper derives and analyzes new diffusion synthetic acceleration (DSA) preconditioners for the $S_N$ transport equation when discretized with a high-order (HO) discontinuous Galerkin (DG) discretization. DSA preconditioners address the need to accelerate the $S_N$ transport equation when the mean free path $\varepsilon$ of particles is small and the condition number of the $S_N$ transport equation scales like $\mathcal{O}( \varepsilon^{-2} )$. By expanding the $S_N$ transport operator in $\varepsilon$ and employing a rigorous singular matrix perturbation analysis, we derive a DSA matrix that reduces to the symmetric interior penalty (SIP) DG discretization of the standard continuum diffusion equation when the mesh is first-order and the total opacity is constant. We prove that preconditioning the HO DG $S_N$ transport equation with the SIP DSA matrix results in an $\mathcal{O}( \varepsilon )$ perturbation of the identity, and fixed-point iteration therefore converges rapidly for optically thick problems. However, the SIP DSA matrix is conditioned like $\mathcal{O}( \varepsilon^{-1} )$, making it difficult to invert for small $\varepsilon$. We further derive a new two-part, additive DSA preconditioner based on a continuous Galerkin discretization of diffusion-reaction, which has a condition number independent of $\varepsilon$, and prove that this DSA variant has the same theoretical efficiency as the SIP DSA preconditioner in the optically thick limit. The analysis is extended to the case of HO (curved) meshes, where so-called mesh cycles can result from elements both being upwind of each other (for a given discrete photon direction). In particular, we prove that performing two additional transport sweeps, with fixed scalar flux, in between DSA steps yields the same theoretical conditioning of fixed-point iterations as in the cycle-free case. Theoretical results are validated by numerical experiments on a HO, highly curved two- and three-dimensional meshes that are generated from an arbitrary Lagrangian--Eulerian hydrodynamics code, where the additional inner sweeps between DSA steps offer up to a 4 x reduction in total number of sweeps required for convergence.

中文翻译：

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高阶弯曲网格上$ S_N $传输的不连续Galerkin离散化的扩散合成加速预处理

SIAM科学计算杂志，第42卷，第5期，第B1271-B1301页，2020年1月。
本文通过高阶（HO）不连续Galerkin（DG）离散化，为$ S_N $输运方程推导并分析了新的扩散合成加速（DSA）预处理器。当粒子的平均自由路径$ \ varepsilon $很小且$ S_N $传输方程的条件数的尺度像$ \ mathcal {O}（\ varepsilon ^ { -2}）$。通过将$ S_N $运输算子扩展为$ \ varepsilon $并采用严格的奇异矩阵摄动分析，我们得出了一个DSA矩阵，该矩阵在第一次使用网格时减少了标准连续谱扩散方程的对称内部罚分（SIP）DG离散化阶，总不透明度是恒定的。我们证明，使用SIP DSA矩阵预处理HO DG $ S_N $传输方程会导致$ \ mathcal {O}（\ varepsilon）$身份扰动，因此定点迭代对于光学较厚的问题迅速收敛。但是，SIP DSA矩阵的条件类似于$ \ mathcal {O}（\ varepsilon ^ {-1}）$，这使得很难为小$ \ varepsilon $求逆。我们进一步基于扩散反应的连续Galerkin离散化推导了一种新的两部分式加性DSA预调节器，其条件数与$ \ varepsilon $无关，并证明此DSA变体与SIP DSA具有相同的理论效率光学厚度范围内的预处理器。分析扩展到HO（弯曲）网格的情况，其中所谓的网格循环可能是由于两个元素彼此上风（对于给定的离散光子方向）而引起的。特别是，我们证明了在DSA步骤之间以固定的标量通量执行两次附加的传输扫描会产生与无循环情况相同的理论上的定点迭代条件。理论结果通过在任意拉格朗日-欧拉流体力学代码生成的HO，高度弯曲的二维和三维网格上进行的数值实验验证，其中DSA步骤之间的附加内部扫描可将总数减少多达4倍收敛所需的扫描次数。在DSA步骤之间进行定点迭代的理论条件与无循环情况相同。理论结果通过在任意拉格朗日-欧拉流体力学代码生成的HO，高度弯曲的二维和三维网格上进行的数值实验验证，其中DSA步骤之间的附加内部扫描可将总数减少多达4倍收敛所需的扫描次数。在DSA步骤之间进行定点迭代的理论条件与无循环情况相同。理论结果通过在任意拉格朗日-欧拉流体力学代码生成的HO，高度弯曲的二维和三维网格上进行的数值实验验证，其中DSA步骤之间的附加内部扫描可将总数减少多达4倍收敛所需的扫描次数。