SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 775-810, April 2024.
Abstract. Cell average decomposition (CAD) plays a critical role in constructing bound-preserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant–Friedrichs–Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss–Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D [math] and [math] spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional [math] and [math] spaces. Yet, it remained unknown for a decade what CAD is optimal for higher-degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carathéodory’s theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D [math] spaces and general 2D [math] spaces of an arbitrary [math]. For the widely used 2D [math] spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi-optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high-order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results.

中文翻译：

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双曲守恒定律高阶保界方案的最优元胞平均分解

SIAM 数值分析杂志，第 62 卷，第 2 期，第 775-810 页，2024 年 4 月。
摘要。元胞平均分解 (CAD) 在构建双曲守恒定律的保界 (BP) 高阶不连续伽辽金法和有限体积法中发挥着关键作用。寻求获得最温和的 BP Courant-Friedrichs-Lewy (CFL) 条件的最佳 CAD (OCAD) 是一个非常重要但又困难的问题。经典的 CAD 是由张和舒于 2010 年使用高斯-洛巴托求积法提出的，在过去十年中得到了广泛的应用。张和舒仅检查了他们的经典 CAD 最佳的一维 [数学] 和 [数学] 空间。然而，我们最近发现经典 CAD 通常对于多维 [数学] 和 [数学] 空间并不是最佳的。然而，十年来，人们仍然不知道哪种 CAD 对于高次多项式空间（尤其是多维空间）来说是最佳的。本文对OCAD问题进行了首次系统分析，建立了一般理论，为设计更高效的BP方案奠定了基础。该分析非常重要，涉及多个数学分支的新技术，包括凸几何中的卡拉西奥多里定理和抽象代数中对称群的不变理论。最值得注意的是，我们发现 OCAD 问题与正多项式圆锥上的正线性函数的多项式优化密切相关，从而建立了四个有用的标准来检查可行 CAD 的最优性。利用已建立的理论，我们严格证明经典 CAD 对于任意 [数学] 的一般一维 [数学] 空间和一般二维 [数学] 空间是最佳的。然而，对于广泛使用的 2D [数学] 空间，经典 CAD 并不是最优的，我们开发了一种通用方法来找出真正的 OCAD，并提出了更实用的准最优 CAD，两者都提供了更温和的 BP CFL与经典 CAD 相比，其条件要求要少得多，但所需的节点要少得多。这些发现显着提高了处理一大类双曲方程的通用高阶 BP 方法的效率，同时只需要对实现代码进行微小的调整。数值结果进一步证实了效率方面的显着优势。