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Very high-order symmetric positive-interior quadrature rules on triangles and tetrahedra
三角形和正四面体上的极高阶对称正内正交规则

Zelalem Arega Worku a , ∗ a , ∗ ^(a,**){ }^{\mathrm{a}, *}, Jason E. Hicken b b ^(b){ }^{\mathrm{b}}, David W. Zingg a a ^(a){ }^{\mathrm{a}} a a ^(a){ }^{a} Institute for Aerospace Studies, University of Toronto, 4925 Dufferin St, Toronto, ON, M3H 5T6, Canada b b ^(b){ }^{b} Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th St, Troy, NY, 2180-3590, USA
b b ^(b){ }^{b} 伦斯勒理工学院机械、航空航天和核工程系，110 8th St, Troy, NY, 2180-3590, USA

Abstract 摘要

We present novel fully-symmetric quadrature rules with positive weights and strictly interior nodes of degrees up to 84 on triangles and 40 on tetrahedra. Initial guesses for solving the nonlinear systems of equations needed to derive quadrature rules are generated by forming tensor-product structures on quadrilateral/hexahedral subdomains of the simplices using the Legendre-Gauss nodes on the first half of the line reference element. In combination with a methodology for node elimination, these initial guesses lead to the development of highly efficient quadrature rules, even for very high polynomial degrees. Using existing estimates of the minimum number of quadrature points for a given degree, we show that the derived quadrature rules on triangles and tetrahedra are more than 95 % 95 % 95%95 \% and 80 % 80 % 80%80 \% efficient, respectively, for almost all degrees. The accuracy of the quadrature rules is demonstrated through numerical examples.
我们提出了新颖的全对称正交规则，这些规则具有正权重和严格的内部节点，在三角形上的节点度数高达 84，在正四面体上的节点度数高达 40。通过使用线参考元素前半部分的 Legendre-Gauss 节点在四边形/六面体子域上形成张量乘积结构，生成求解正交规则所需的非线性方程组的初始猜测。结合消除节点的方法，这些初始猜测可以开发出高效的正交规则，即使对于非常高的多项式度数也是如此。利用现有的对给定阶数的最小正交点数的估计，我们证明了在三角形和正四面体上推导出的正交规则在几乎所有阶数上的效率分别超过 95 % 95 % 95%95 \% 和 80 % 80 % 80%80 \% 。正交规则的准确性将通过数值示例来证明。

Keywords: Positive-interior quadrature, Gaussian quadrature, Numerical integration, Triangle, Tetrahedron
关键词正内部正交 高斯正交 数值积分 三角形 四面体
2020 MSC: 65D32, 65M06, 65M60
2020 MSC：65D32、65M06、65M60

1. Introduction 1.导言

Quadrature rules are indispensible tools in numerical methods for solving partial differential equations (PDEs), providing an efficient means of approximating integrals. Often numerical discretizations of problems involving complex geometries use simplicial meshes, necessitating the development of quadrature rules on triangles and tetrahedra. While several studies have proposed efficient high-order quadrature rules on these elements, e.g., [1, 2, 3, 4, 5, 6], futher improvements in terms of polynomial order and efficiency remain potentially beneficial. Currently, the highest degree symmetric quadrature rules with positive weights and interior nodes (PI) on triangles and tetrahedra are 50 [3] and 20 [ 5 , 6 ] 20 [ 5 , 6 ] 20[5,6]20[5,6], respectively. These accuracy levels, however, may not be sufficiently high in scenarios where a degree p p pp finiteelement or discontinuous Galerkin (DG) discretization employs very high-order quadrature
正交规则是求解偏微分方程（PDEs）的数值方法中不可或缺的工具，提供了近似积分的有效方法。对涉及复杂几何图形的问题进行数值离散时，通常会使用简单网格，这就需要开发三角形和正四面体的正交规则。虽然已有一些研究提出了针对这些元素的高效高阶正交规则，例如 [1, 2, 3, 4, 5, 6]，但在多项式阶数和效率方面的进一步改进仍有潜在益处。目前，三角形和正四面体上具有正权重和内部节点 (PI) 的最高度对称正交规则分别为 50 [3] 和 20 [ 5 , 6 ] 20 [ 5 , 6 ] 20[5,6]20[5,6] 。然而，在有限元或非连续加勒金 (DG) 离散化采用非常高阶正交的情况下，这些精度水平可能不够高。
rules, e.g., 3 p 3 p 3p3 p up to 6 p [ 7 , 8 , 9 ] 6 p [ 7 , 8 , 9 ] 6p[7,8,9]6 p[7,8,9], for accuracy or stability purposes. In general, the development of very high-order discretization operators is limited by the availability of sufficiently accurate quadrature rules. For example, the construction of summation-by-parts (SBP) operators of degree p p pp, which are important for the design of provably stable discretizations of PDEs [10, 11], requires at least degree 2 p − 1 2 p − 1 2p-12 p-1 quadrature rules [12, 13]; consequently, these operators can currently be constructed only up to degree 10 on tetrahedra. The development of very high-order quadrature rules can significantly improve the design of numerical methods for PDEs, with their impact potentially extending to the many other fields where quadrature rules are applied.
规则，例如 3 p 3 p 3p3 p 直至 6 p [ 7 , 8 , 9 ] 6 p [ 7 , 8 , 9 ] 6p[7,8,9]6 p[7,8,9] ，以达到精确或稳定的目的。一般来说，高阶离散化算子的发展受到足够精确的正交规则的限制。例如，度数为 p p pp 的逐段求和 (SBP) 算子的构造对于设计可证明稳定的 PDE 离散化非常重要 [10, 11]，它至少需要度数为 2 p − 1 2 p − 1 2p-12 p-1 的正交规则 [12, 13]；因此，这些算子目前只能在四面体上构造到 10 度。开发极高阶正交规则可以显著改进 PDE 数值方法的设计，其影响可能扩展到应用正交规则的许多其他领域。
The properties of the quadrature rule used in a numerical method can affect the accuracy, efficiency, and stability of the entire simulation. Symmetry of the quadrature rules ensures that no unphysical asymmetry is introduced to a problem of interest [14, 15]. Furthermore, symmetric nodes lead to difference operators with inherent symmetry, which can be exploited to reduce memory requirements. Positivity of the weights guarantees that the quadrature approximations of integrals of positive functions remain positive and prevents errors due to cancellations of quadrature contributions at each node. Furthermore, positivity of the weights is essential in the design of stable numerical methods, for example in the context of SBP discretizations, and it produces a valid norm that can be used to bound the energy or entropy functions associated with PDEs. Finally, the constraint that all quadrature nodes must be strictly inside the domain can be essential for practical implementations and approximation of integrals of functions with singularities at element facets.
数值方法中使用的正交规则的特性会影响整个模拟的精度、效率和稳定性。正交规则的对称性可确保相关问题不会引入非物理的不对称[14, 15]。此外，对称节点还能产生具有内在对称性的差分算子，从而降低内存需求。权重的正向性保证了正函数积分的正交近似值保持为正，并防止了由于每个节点的正交贡献抵消而产生的误差。此外，权重的正向性对于设计稳定的数值方法至关重要，例如在 SBP 离散化的情况下，权重的正向性产生的有效规范可用于约束与 PDE 相关的能量或熵函数。最后，所有正交节点必须严格位于域内这一约束对于实际应用和近似元素面上具有奇点的函数积分至关重要。
Gaussian-type PI quadrature rules have been studied extensively, and it is well-known that Gaussian quadrature rules are optimal in one dimension [16]. In some cases, quadrature rules with a subset of nodes placed at the facets, such as those presented in [17], can be more efficient for numerical discretizations of PDEs despite having more nodes than PI quadrature rules. In many cases, however, the small number of nodes of PI quadrature rules translates into higher efficiency. While very efficient symmetric PI rules, up to degree 50, have been constructed on triangles [3], the highest available degree on the tetrahedron has been limited to 15 [3, 2] until the recent extension up to degree 20 [ 5 , 6 ] 20 [ 5 , 6 ] 20[5,6]20[5,6]. This is because the construction of quadrature rules in three dimensions is notoriously difficult for large values of p [ 18 , 19 ] p [ 18 , 19 ] p[18,19]p[18,19]. One of the primary challenges arises from the absence of goodquality initial guesses, which are crucial for the convergence of nonlinear equation solvers, particularly as the polynomial degree increases. In this paper, we present a novel approach for generating initial guesses to solve the nonlinear equations needed to derive quadrature rules. This approach involves forming tensor-product structures on quadrilateral/hexahedral subdomains of the triangular/tetrahedral reference elements using the Legendre-Gauss (LG) nodes on the first half of the line reference element. The initial guesses, coupled with a methodology for node elimination, are used to derive efficient fully-symmetric PI quadrature rules of degree up to 84 on triangles and 40 on tetrahedra.
高斯型 PI 正交规则已被广泛研究，众所周知，高斯正交规则在一维上是最优的 [16]。在某些情况下，在面上放置节点子集的正交规则（如文献[17]中提出的正交规则），尽管比 PI 正交规则有更多的节点，但对于 PDE 的数值离散化却更有效。然而，在许多情况下，PI 正交规则的节点数少却能带来更高的效率。虽然在三角形上已经构建了高达 50 度的高效对称 PI 正交规则 [3]，但在最近扩展到 20 [ 5 , 6 ] 20 [ 5 , 6 ] 20[5,6]20[5,6] 度之前，四面体上可用的最高度数一直限于 15 [3, 2]。这是因为对于 p [ 18 , 19 ] p [ 18 , 19 ] p[18,19]p[18,19] 的大值，在三维空间构建正交规则是出了名的困难。主要挑战之一是缺乏高质量的初始猜测，而初始猜测对于非线性方程求解器的收敛至关重要，尤其是当多项式度数增加时。在本文中，我们提出了一种生成初始猜测的新方法，用于求解推导正交规则所需的非线性方程。这种方法是在三角形/四面体参考元素的四边形/六面体子域上，利用线参考元素前半部分的 Legendre-Gauss (LG) 节点形成张量乘积结构。初始猜测与节点消除方法相结合，可用于推导出高效的全对称 PI 正交规则，在三角形上的阶数最高可达 84，在正四面体上的阶数最高可达 40。
The remaining sections are organized as follows. Section 2 presents the symmetry orbits on the reference triangle and tetrahedron, the systems of nonlinear equations that need to be solved, and some lower-bound estimates of Gaussian-type quadrature rules from the literature. Section 3 delineates the general algorithm for deriving the quadrature rules and the method of solution. The initial guess generation is presented in Section 4, which is followed by a discussion on the derivation and efficiency of the line-Legendre-Gauss (line-LG)
其余章节安排如下。第 2 节介绍了参考三角形和四面体上的对称轨道、需要求解的非线性方程组，以及文献中对高斯型正交规则的一些下限估计。第 3 节阐述了推导正交规则的一般算法和求解方法。第 4 节介绍了初始猜测的生成，随后讨论了线-勒根德雷-高斯（line-LG）正交规则的推导和效率。
quadrature rules in Section 5. Section 6 presents a node elimination strategy. A discussion on the derived novel quadrature rules is provided in Section 7. Finally, some numerical results are presented in Section 8, followed by conclusions in Section 9.
第 5 节介绍正交规则。第 6 节介绍了节点消除策略。第 7 节讨论了推导出的新正交规则。最后，第 8 节给出了一些数值结果，第 9 节给出了结论。

2. Preliminaries and Background
2.前言和背景

For the construction of the quadrature rules in this work, we consider the triangle and tetrahedron reference elements defined, respectively, as
为了构建正交规则，我们将三角形和四面体参考元素分别定义为
Ω tri = { ( x 1 , x 2 ) ∣ x 1 , x 2 ≥ − 1 ; x 1 + x 2 ≤ 0 } Ω tet = { ( x 1 , x 2 , x 3 ) ∣ x 1 , x 2 , x 3 ≥ − 1 ; x 1 + x 2 + x 3 ≤ − 1 } Ω tri = x 1 , x 2 ∣ x 1 , x 2 ≥ − 1 ; x 1 + x 2 ≤ 0 Ω tet = x 1 , x 2 , x 3 ∣ x 1 , x 2 , x 3 ≥ − 1 ; x 1 + x 2 + x 3 ≤ − 1 {:[Omega_(tri)={(x_(1),x_(2))∣x_(1),x_(2) >= -1;x_(1)+x_(2) <= 0}],[Omega_(tet)={(x_(1),x_(2),x_(3))∣x_(1),x_(2),x_(3) >= -1;x_(1)+x_(2)+x_(3) <= -1}]:}\begin{aligned} & \Omega_{\mathrm{tri}}=\left\{\left(x_{1}, x_{2}\right) \mid x_{1}, x_{2} \geq-1 ; x_{1}+x_{2} \leq 0\right\} \\ & \Omega_{\mathrm{tet}}=\left\{\left(x_{1}, x_{2}, x_{3}\right) \mid x_{1}, x_{2}, x_{3} \geq-1 ; x_{1}+x_{2}+x_{3} \leq-1\right\} \end{aligned}
where x = [ x 1 , … , x d ] T x = x 1 , … , x d T x=[x_(1),dots,x_(d)]^(T)\boldsymbol{x}=\left[x_{1}, \ldots, x_{d}\right]^{T} denotes the Cartesian coordinates on the reference elements. The line reference element is used in the initial guess generation procedure and is defined as Ω line = { x 1 ∣ − 1 ≤ x 1 ≤ 1 } Ω line  = x 1 ∣ − 1 ≤ x 1 ≤ 1 Omega_("line ")={x_(1)∣-1 <= x_(1) <= 1}\Omega_{\text {line }}=\left\{x_{1} \mid-1 \leq x_{1} \leq 1\right\}.
其中 x = [ x 1 , … , x d ] T x = x 1 , … , x d T x=[x_(1),dots,x_(d)]^(T)\boldsymbol{x}=\left[x_{1}, \ldots, x_{d}\right]^{T} 表示参考元素的直角坐标。线参考元素用于初始猜测生成程序，定义为 Ω line = { x 1 ∣ − 1 ≤ x 1 ≤ 1 } Ω line  = x 1 ∣ − 1 ≤ x 1 ≤ 1 Omega_("line ")={x_(1)∣-1 <= x_(1) <= 1}\Omega_{\text {line }}=\left\{x_{1} \mid-1 \leq x_{1} \leq 1\right\} 。
There are three symmetry groups on the triangle, and five on the tetrahedron [20, 2, 4]. On the triangle, the symmetry groups, in barycentric coordinates, are permutations of
三角形上有三个对称组，四面体上有五个对称组 [20, 2, 4]。在三角形上，以重心坐标表示的对称组是
S 1 = ( 1 3 , 1 3 , 1 3 ) , S 21 = ( α , α , 1 − 2 α ) , S 111 = ( α , β , 1 − α − β ) , S 1 = 1 3 , 1 3 , 1 3 , S 21 = ( α , α , 1 − 2 α ) , S 111 = ( α , β , 1 − α − β ) , S_(1)=((1)/(3),(1)/(3),(1)/(3)),quadS_(21)=(alpha,alpha,1-2alpha),quadS_(111)=(alpha,beta,1-alpha-beta),S_{1}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right), \quad S_{21}=(\alpha, \alpha, 1-2 \alpha), \quad S_{111}=(\alpha, \beta, 1-\alpha-\beta),
where α α alpha\alpha and β β beta\beta are parameters such that the quadrature points lie strictly in the interior of the domain. Similarly, the symmetry groups on the reference tetrahedron are permutations of
其中 α α alpha\alpha 和 β β beta\beta 为参数，使得正交点严格位于域的内部。同样，参照四面体上的对称组是
S 1 = ( 1 4 , 1 4 , 1 4 , 1 4 ) , S 31 = ( α , α , α , 1 − 3 α ) , S 22 = ( α , α , 1 2 − α , 1 2 − α ) , S 211 = ( α , α , β , 1 − 2 α − β ) , S 1111 = ( α , β , γ , 1 − α − β − γ ) . S 1 = 1 4 , 1 4 , 1 4 , 1 4 , S 31 = ( α , α , α , 1 − 3 α ) , S 22 = α , α , 1 2 − α , 1 2 − α , S 211 = ( α , α , β , 1 − 2 α − β ) , S 1111 = ( α , β , γ , 1 − α − β − γ ) . {:[S_(1),=((1)/(4),(1)/(4),(1)/(4),(1)/(4))",",S_(31),=(alpha","alpha","alpha","1-3alpha)","quadS_(22)=(alpha,alpha,(1)/(2)-alpha,(1)/(2)-alpha)","],[S_(211),=(alpha","alpha","beta","1-2alpha-beta)","quadS_(1111),=(alpha","beta","gamma","1-alpha-beta-gamma).]:}\begin{array}{rlrl} S_{1} & =\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right), & S_{31} & =(\alpha, \alpha, \alpha, 1-3 \alpha), \quad S_{22}=\left(\alpha, \alpha, \frac{1}{2}-\alpha, \frac{1}{2}-\alpha\right), \\ S_{211} & =(\alpha, \alpha, \beta, 1-2 \alpha-\beta), \quad S_{1111} & =(\alpha, \beta, \gamma, 1-\alpha-\beta-\gamma) . \end{array}
The problem statement for deriving a quadrature rule on the domain Ω Ω Omega\Omega can be posed as: find y y y\boldsymbol{y} and w w w\boldsymbol{w} such that
在域 Ω Ω Omega\Omega 上推导正交规则的问题陈述可表述为：求 y y y\boldsymbol{y} 和 w w w\boldsymbol{w} ，使得
∫ Ω P j ( x ) d Ω = ∑ i = 1 n q w i P j ( y i ) , j ∈ { 1 , … , n b } ∫ Ω   P j ( x ) d Ω = ∑ i = 1 n q   w i P j y i , j ∈ 1 , … , n b int_(Omega)P_(j)(x)dOmega=sum_(i=1)^(n_(q))w_(i)P_(j)(y_(i)),quad j in{1,dots,n_(b)}\int_{\Omega} \mathcal{P}_{j}(\boldsymbol{x}) \mathrm{d} \Omega=\sum_{i=1}^{n_{q}} w_{i} \mathcal{P}_{j}\left(\boldsymbol{y}_{i}\right), \quad j \in\left\{1, \ldots, n_{b}\right\}
where y y y\boldsymbol{y} is the vector of the coordinate tuples of the nodes, w w w\boldsymbol{w} is the vector of quadrature weights, and n q n q n_(q)n_{q} and n b n b n_(b)n_{b} denote the number of quadrature points and polynomial basis functions, respectively. For a degree q q qq accurate quadrature rule on a simplex, there are n b = ( q + d d ) n b = ( q + d d ) n_(b)=((q+d)/(d))n_{b}=\binom{q+d}{d} polynomial basis functions. Rewriting the problem statement in a matrix form, we have
其中， y y y\boldsymbol{y} 是节点坐标元组向量， w w w\boldsymbol{w} 是正交权向量， n q n q n_(q)n_{q} 和 n b n b n_(b)n_{b} 分别表示正交点数和多项式基函数数。对于单纯形上的 q q qq 度精确正交规则，有 n b = ( q + d d ) n b = ( q + d d ) n_(b)=((q+d)/(d))n_{b}=\binom{q+d}{d} 个多项式基函数。将问题陈述改写成矩阵形式，我们可以得到
g := V T w − f = 0 g := V T w − f = 0 g:=V^(T)w-f=0\boldsymbol{g}:=\mathrm{V}^{T} \boldsymbol{w}-\boldsymbol{f}=\mathbf{0}
where V is the Vandermonde matrix containing evaluations of the basis functions at each node along its columns and f = [ ∫ Ω P 1 ( x ) d Ω , … , ∫ Ω P n b ( x ) d Ω ] T f = ∫ Ω   P 1 ( x ) d Ω , … , ∫ Ω   P n b ( x ) d Ω T f=[int_(Omega)P_(1)(x)dOmega,dots,int_(Omega)P_(n_(b))(x)dOmega]^(T)\boldsymbol{f}=\left[\int_{\Omega} \mathcal{P}_{1}(\boldsymbol{x}) \mathrm{d} \Omega, \ldots, \int_{\Omega} \mathcal{P}_{n_{b}}(\boldsymbol{x}) \mathrm{d} \Omega\right]^{T}. We use the orthonormal Proriol-Koornwinder-Dubiner (PKD) [21, 22, 23] basis functions, which produce wellconditioned Vandermonde matrices and are convenient, as all except the first entry of f f f\boldsymbol{f} are zero due to the orthogonality of the basis functions.
其中，V 是范德蒙德矩阵，包含基函数在每个节点沿其列的求值， f = [ ∫ Ω P 1 ( x ) d Ω , … , ∫ Ω P n b ( x ) d Ω ] T f = ∫ Ω   P 1 ( x ) d Ω , … , ∫ Ω   P n b ( x ) d Ω T f=[int_(Omega)P_(1)(x)dOmega,dots,int_(Omega)P_(n_(b))(x)dOmega]^(T)\boldsymbol{f}=\left[\int_{\Omega} \mathcal{P}_{1}(\boldsymbol{x}) \mathrm{d} \Omega, \ldots, \int_{\Omega} \mathcal{P}_{n_{b}}(\boldsymbol{x}) \mathrm{d} \Omega\right]^{T} 。我们使用正交的 Proriol-Koornwinder-Dubiner (PKD) [21, 22, 23] 基函数，它能产生条件良好的范德蒙德矩阵，而且由于基函数的正交性，除 f f f\boldsymbol{f} 的第一个条目外，其他条目均为零，因此非常方便。

2.1. Lower-Bound Estimates for the PI Quadrature Rules
2.1.PI 正交规则的下限估计值

It is well known that the one-dimensional LG quadrature rules are optimal in the sense that they use the smallest possible number of nodes to integrate polynomials exactly [16]. Such optimal rules in higher dimensions are not known, but there are some results that estimate the lower bounds of the number of nodes required to integrate polynomials exactly. In this work, we are interested not only in the estimates of the total number of nodes but also in the number of nodes in each symmetry orbit, which serves as a guide for node elimination. For the triangle, we use the estimates of Lyness and Jespersen [24], which can be stated as follows. Let α s = [ 3 , − 4 , − 1 , 0 , − 1 , − 4 ] , α q = α s [ ( q mod 6 ) + 1 ] α s = [ 3 , − 4 , − 1 , 0 , − 1 , − 4 ] , α q = α s [ ( q mod 6 ) + 1 ] alpha_(s)=[3,-4,-1,0,-1,-4],alpha_(q)=alpha_(s)[(q mod6)+1]\alpha_{s}=[3,-4,-1,0,-1,-4], \alpha_{q}=\alpha_{s}[(q \bmod 6)+1], and E ( q , α q ) = ( ( q + 3 ) 2 + α q ) / 12 E q , α q = ( q + 3 ) 2 + α q / 12 E(q,alpha_(q))=((q+3)^(2)+alpha_(q))//12\mathcal{E}\left(q, \alpha_{q}\right)=\left((q+3)^{2}+\alpha_{q}\right) / 12; then the lower-bound estimate for each symmetry group is given by
众所周知，一维 LG 正交规则是最优规则，即使用尽可能少的节点数精确积分多项式 [16]。在更高维度上，这种最优规则尚不为人所知，但有一些结果估计了精确积分多项式所需的节点数下限。在这项工作中，我们不仅对总节点数的估计感兴趣，而且对每个对称轨道中的节点数也感兴趣，这可以作为消除节点的指南。对于三角形，我们使用 Lyness 和 Jespersen [24] 的估计值，可表述如下。设 α s = [ 3 , − 4 , − 1 , 0 , − 1 , − 4 ] , α q = α s [ ( q mod 6 ) + 1 ] α s = [ 3 , − 4 , − 1 , 0 , − 1 , − 4 ] , α q = α s [ ( q mod 6 ) + 1 ] alpha_(s)=[3,-4,-1,0,-1,-4],alpha_(q)=alpha_(s)[(q mod6)+1]\alpha_{s}=[3,-4,-1,0,-1,-4], \alpha_{q}=\alpha_{s}[(q \bmod 6)+1] 和 E ( q , α q ) = ( ( q + 3 ) 2 + α q ) / 12 E q , α q = ( q + 3 ) 2 + α q / 12 E(q,alpha_(q))=((q+3)^(2)+alpha_(q))//12\mathcal{E}\left(q, \alpha_{q}\right)=\left((q+3)^{2}+\alpha_{q}\right) / 12 ；则每个对称组的下限估计值为
| S 111 | = { 0 , q < 6 ⌊ E ( q − 6 , α q ) + 2 3 ⌋ , q ≥ 6 , | S 12 | = ⌊ E ( q , α q ) − 3 S 111 2 ⌋ , | S 1 | = { 0 , 1 + 2 S 12 + 3 S 111 > E ( q , α q ) , 1 , otherwise , S 111 = 0 , q < 6 E q − 6 , α q + 2 3 , q ≥ 6 , S 12 = E q , α q − 3 S 111 2 , S 1 = 0 , 1 + 2 S 12 + 3 S 111 > E q , α q , 1 ,  otherwise  , {:[|S_(111)|={[0",",q < 6],[|__(E(q-6,alpha_(q))+2)/(3)__|",",q >= 6],quad|S_(12)|=|__(E(q,alpha_(q))-3S_(111))/(2)__|,:}],[|S_(1)|={[0",",1+2S_(12)+3S_(111) > E(q,alpha_(q))","],[1","," otherwise "","]:}]:}\begin{aligned} \left|S_{111}\right| & =\left\{\begin{array}{ll} 0, & q<6 \\ \left\lfloor\frac{\mathcal{E}\left(q-6, \alpha_{q}\right)+2}{3}\right\rfloor, & q \geq 6 \end{array}, \quad\left|S_{12}\right|=\left\lfloor\frac{\mathcal{E}\left(q, \alpha_{q}\right)-3 S_{111}}{2}\right\rfloor,\right. \\ \left|S_{1}\right| & = \begin{cases}0, & 1+2 S_{12}+3 S_{111}>\mathcal{E}\left(q, \alpha_{q}\right), \\ 1, & \text { otherwise },\end{cases} \end{aligned}
where ⌊ ⋅ ⌋ ⌊ ⋅ ⌋ |__*__|\lfloor\cdot\rfloor indicates the floor operator. The total number of nodes on the triangle is computed as
其中 ⌊ ⋅ ⌋ ⌊ ⋅ ⌋ |__*__|\lfloor\cdot\rfloor 表示下限算子。三角形上的节点总数计算公式为
n q = | S 1 | + 3 | S 21 | + 6 | S 111 | n q = S 1 + 3 S 21 + 6 S 111 n_(q)=|S_(1)|+3|S_(21)|+6|S_(111)|n_{q}=\left|S_{1}\right|+3\left|S_{21}\right|+6\left|S_{111}\right|
For the tetrahedron, we use the estimates of Wang and Papanicolopulos [25], which can be stated as follows. Let I ( q , a ) I ( q , a ) I(q,a)\mathcal{I}(q, a) be the function
对于四面体，我们使用 Wang 和 Papanicolopulos [25] 的估计值，具体如下。设 I ( q , a ) I ( q , a ) I(q,a)\mathcal{I}(q, a) 为函数
I ( q , a ) = { 1 , q ≥ a 0 , otherwise I ( q , a ) = 1 ,      q ≥ a 0 ,       otherwise  I(q,a)={[1",",q >= a],[0","," otherwise "]:}\mathcal{I}(q, a)= \begin{cases}1, & q \geq a \\ 0, & \text { otherwise }\end{cases}
and define the following coefficients
并定义以下系数
q 12 = q − 12 , m 2 = ⌊ q 2 − 1 ⌋ I ( q , 4 ) , m 3 = ⌊ ( q 2 − 2 ) 2 ⌋ I ( q , 6 ) m 4 = ⌊ ( q 12 + 4 ) 3 + 3 ( q 12 + 4 ) 2 − 9 ( q 12 + 4 ) ( ( q 12 + 4 ) mod 2 ) 144 ⌋ I ( q 12 , 0 ) , m e = ⌊ ( q + 4 ) 3 + 3 ( q + 4 ) 2 − 9 ( q + 4 ) ( ( q + 4 ) mod 2 ) 144 ⌋ I ( q , 0 ) q 12 = q − 12 , m 2 = q 2 − 1 I ( q , 4 ) , m 3 = q 2 − 2 2 I ( q , 6 ) m 4 = q 12 + 4 3 + 3 q 12 + 4 2 − 9 q 12 + 4 q 12 + 4 mod 2 144 I q 12 , 0 , m e = ( q + 4 ) 3 + 3 ( q + 4 ) 2 − 9 ( q + 4 ) ( ( q + 4 ) mod 2 ) 144 I ( q , 0 ) {:[q_(12)=q-12","quadm_(2)=|__(q)/(2)-1__|I(q","4)","quadm_(3)=|__((q)/(2)-2)^(2)__|I(q","6)],[m_(4)=|__((q_(12)+4)^(3)+3(q_(12)+4)^(2)-9(q_(12)+4)((q_(12)+4)mod2))/(144)__|I(q_(12),0)","],[m_(e)=|__((q+4)^(3)+3(q+4)^(2)-9(q+4)((q+4)mod2))/(144)__|I(q","0)]:}\begin{aligned} & q_{12}=q-12, \quad m_{2}=\left\lfloor\frac{q}{2}-1\right\rfloor \mathcal{I}(q, 4), \quad m_{3}=\left\lfloor\left(\frac{q}{2}-2\right)^{2}\right\rfloor \mathcal{I}(q, 6) \\ & m_{4}=\left\lfloor\frac{\left(q_{12}+4\right)^{3}+3\left(q_{12}+4\right)^{2}-9\left(q_{12}+4\right)\left(\left(q_{12}+4\right) \bmod 2\right)}{144}\right\rfloor \mathcal{I}\left(q_{12}, 0\right), \\ & m_{e}=\left\lfloor\frac{(q+4)^{3}+3(q+4)^{2}-9(q+4)((q+4) \bmod 2)}{144}\right\rfloor \mathcal{I}(q, 0) \end{aligned}
where ⌊ ⋅ ⌉ ⌊ ⋅ ⌉ |__*~|\lfloor\cdot\rceil denotes rounding to the closest integer. Then a quadrature rule on the tetrahedron with the following number of nodes in each symmetry group provides a lower-bound estimate of the number of nodes required to satisfy a degree q q qq quadrature accuracy,
其中 ⌊ ⋅ ⌉ ⌊ ⋅ ⌉ |__*~|\lfloor\cdot\rceil 表示四舍五入到最接近的整数。然后，在每个对称组的节点数如下的四面体上使用正交规则，就可以得到满足度 q q qq 正交精度所需的节点数的下限估计值、
| S 1111 | = ⌈ m 4 4 ⌉ , | S 211 | = ⌈ m 4 + m 3 − 4 S 1111 3 ⌉ , | S 22 | = ⌈ m 4 + m 3 + m 2 − 3 S 211 − 4 S 1111 2 ⌉ , | S 31 | = ⌊ m e − 2 S 22 − 3 S 211 − 4 S 1111 2 ⌋ , | S 1 | = m e − 2 S 31 − 2 S 22 − 3 S 211 − 4 S 1111 S 1111 = m 4 4 , S 211 = m 4 + m 3 − 4 S 1111 3 , S 22 = m 4 + m 3 + m 2 − 3 S 211 − 4 S 1111 2 , S 31 = m e − 2 S 22 − 3 S 211 − 4 S 1111 2 , S 1 = m e − 2 S 31 − 2 S 22 − 3 S 211 − 4 S 1111 {:[|S_(1111)|=|~(m_(4))/(4)~|","quad|S_(211)|=|~(m_(4)+m_(3)-4S_(1111))/(3)~|","quad|S_(22)|=|~(m_(4)+m_(3)+m_(2)-3S_(211)-4S_(1111))/(2)~|","],[|S_(31)|=|__(m_(e)-2S_(22)-3S_(211)-4S_(1111))/(2)__|","quad|S_(1)|=m_(e)-2S_(31)-2S_(22)-3S_(211)-4S_(1111)]:}\begin{aligned} \left|S_{1111}\right| & =\left\lceil\frac{m_{4}}{4}\right\rceil, \quad\left|S_{211}\right|=\left\lceil\frac{m_{4}+m_{3}-4 S_{1111}}{3}\right\rceil, \quad\left|S_{22}\right|=\left\lceil\frac{m_{4}+m_{3}+m_{2}-3 S_{211}-4 S_{1111}}{2}\right\rceil, \\ \left|S_{31}\right| & =\left\lfloor\frac{m_{e}-2 S_{22}-3 S_{211}-4 S_{1111}}{2}\right\rfloor, \quad\left|S_{1}\right|=m_{e}-2 S_{31}-2 S_{22}-3 S_{211}-4 S_{1111} \end{aligned}
where ⌈ ⋅ ⌉ ⌈ ⋅ ⌉ |~*~|\lceil\cdot\rceil denotes the ceiling operator. The total number of nodes on the tetrahedron is computed as
其中 ⌈ ⋅ ⌉ ⌈ ⋅ ⌉ |~*~|\lceil\cdot\rceil 表示上限算子。四面体上的节点总数计算公式为
n q = | S 1 | + 4 | S 31 | + 6 | S 22 | + 12 | S 211 | + 24 | S 1111 | n q = S 1 + 4 S 31 + 6 S 22 + 12 S 211 + 24 S 1111 n_(q)=|S_(1)|+4|S_(31)|+6|S_(22)|+12|S_(211)|+24|S_(1111)|n_{q}=\left|S_{1}\right|+4\left|S_{31}\right|+6\left|S_{22}\right|+12\left|S_{211}\right|+24\left|S_{1111}\right|
We note that the lower-bound estimates stated above may not produce optimal quadrature rules as the assumptions used to derive them are not necessarily true for all cases. Furthermore, the quadrature rules satisfying the estimates may require placing nodes outside the domain or using negative weights; we refer interested readers to [24] and [25] for further details. With these caveats aside, we use these lower-bound estimates to define efficiency of the PI quadrature rules conservatively as
我们注意到，上述下限估计值可能不会产生最优的正交规则，因为用于推导这些估计值的假设并不一定适用于所有情况。此外，满足估计值的正交规则可能需要将节点置于域外或使用负权重；更多详情请读者参阅 [24] 和 [25]。撇开这些注意事项不谈，我们使用这些下限估计值，保守地定义 PI 正交规则的效率为
e q := n q m n q e q := n q m n q e_(q):=(n_(q)^(m))/(n_(q))e_{q}:=\frac{n_{q}^{m}}{n_{q}}
where n q n q n_(q)n_{q} is the number of nodes of a given degree q q qq quadrature rule and n q m n q m n_(q)^(m)n_{q}^{m} is the number of nodes given by the lower-bound estimates stated in (7) for triangles and (11) for tetrahedra.
其中， n q n q n_(q)n_{q} 是给定度数 q q qq 正交规则的节点数， n q m n q m n_(q)^(m)n_{q}^{m} 是根据三角形的下限估计值 (7) 和四面体的下限估计值 (11) 得出的节点数。

3. Methodology 3.方法论

The first step in solving the quadrature problem, (6), is generating some intial guesses for the quadrature points and weights, which are used to compute the Vandermonde matrix and weight vector. In general, the initial guesses do not satisfy (6), and will be used to iteratively reduce the residual error to machine precision. Generating initial guesses that fall within the radius of convergence of the (gradient-based) iterative solver is a challenging task and a key factor limiting the derivation of very high-order quadrature rules on tetrahedra.
解决正交问题 (6) 的第一步是生成一些正交点和权重的初始猜测，用于计算范德蒙德矩阵和权重向量。一般情况下，初始猜测不满足 (6)，将用于迭代降低残余误差至机器精度。生成收敛半径在（基于梯度的）迭代求解器收敛半径范围内的初始猜测是一项具有挑战性的任务，也是限制推导四面体高阶正交规则的关键因素。
The methodology employed in this study can be summarized as follows. First, we introduce an efficient approach to generate initial guesses that is based on mapping the LG quadrature nodes in the first half of the line reference element onto specific lines in the simplex. Next, we set (6) as a minimization problem and solve it using a gradient-based iterative solver to obtain intermediate quadrature rules which we refer to as line-LG quadrature rules. Finally, we apply a node elimination technique to the line-LG rules and derive novel quadrature rules with improved efficiency. The quadrature rules obtained after node elimination will be referred to as “new” in tables and figures. Further details of the algorithm are illustrated in Fig. 1.
本研究采用的方法可归纳如下。首先，我们引入了一种生成初始猜测的有效方法，该方法基于将线参考元素前半部分的 LG 正交节点映射到单纯形中的特定线上。接下来，我们将 (6) 设置为最小化问题，并使用基于梯度的迭代求解器进行求解，从而获得中间正交规则，我们称之为线-LG 正交规则。最后，我们在线性-线性正交规则中应用节点消除技术，得出效率更高的新型正交规则。消除节点后得到的正交规则在表格和图表中称为 "新 "规则。图 1 展示了算法的更多细节。
The method of solution used in this work is similar to that described in [17] for quadrature rules with a subset of nodes placed at the facets for the construction of SBP diagonal-E multidimensional SBP operators [26, 12]. In this work, the approach is applied to derive quadrature rules with strictly interior nodes, using only a gradient-based iterative solver instead of the coupled gradient and stochastic-based solver employed in [17]. For completeness, we describe the approach with some adjustments to accommodate the aforementioned differences.
本研究采用的求解方法类似于 [17] 中描述的正交规则，即在 SBP 对角线-E 多维 SBP 算子[26, 12]的构造中，在面上放置节点子集[26, 12]。在本研究中，我们只使用基于梯度的迭代求解器，而不是 [17] 中使用的基于梯度和随机的耦合求解器，来推导具有严格内部节点的正交规则。为完整起见，我们对该方法进行了一些调整，以适应上述差异。
Using the initial guesses, it is possible to compute the coordinates of the i i ii-th node by applying the transformation,
利用初始猜测，可以通过应用变换来计算 i i ii -th 节点的坐标、
y i = T T λ k ( i ) y i = T T λ k ( i ) y_(i)=T^(T)lambda_(k)^((i))\boldsymbol{y}_{i}=\mathrm{T}^{T} \boldsymbol{\lambda}_{k}^{(i)}
where T ∈ R ( d + 1 ) × d T ∈ R ( d + 1 ) × d TinR^((d+1)xx d)\mathrm{T} \in \mathbb{R}^{(d+1) \times d} contains the coordinates of the d + 1 d + 1 d+1d+1 vertices in its rows and λ k ( i ) λ k ( i ) lambda_(k)^((i))\boldsymbol{\lambda}_{k}^{(i)} is the k k kk-th permutation of the barycentric coordinates of the symmetry group that corresponds to
其中， T ∈ R ( d + 1 ) × d T ∈ R ( d + 1 ) × d TinR^((d+1)xx d)\mathrm{T} \in \mathbb{R}^{(d+1) \times d} 包含其行中 d + 1 d + 1 d+1d+1 顶点的坐标， λ k ( i ) λ k ( i ) lambda_(k)^((i))\boldsymbol{\lambda}_{k}^{(i)} 是对称组偏心坐标的第 k k kk 次排列，对应于
Figure 1: Algorithm flowchart for generating symmetric PI quadrature rules.
图 1：生成对称 PI 正交规则的算法流程图。
the i i ii-th node. The weight vector, w w w\boldsymbol{w}, is constructed by assigning equal weights to all nodes in the same symmetry group.
第 i i ii 个节点。权重向量 w w w\boldsymbol{w} 是通过为同一对称组中的所有节点分配相等的权重而构建的。
We note that the problem in (6) can be written equivalently as a minimization problem,
我们注意到，（6）中的问题可以等同于最小化问题、
min τ 1 2 g T g min τ   1 2 g T g min_(tau)(1)/(2)g^(T)g\min _{\boldsymbol{\tau}} \frac{1}{2} \boldsymbol{g}^{T} \boldsymbol{g}
where τ = [ λ ^ T , w ^ T ] T τ = λ ^ T , w ^ T T tau=[ widehat(lambda)^(T), widehat(w)^(T)]^(T)\boldsymbol{\tau}=\left[\widehat{\boldsymbol{\lambda}}^{T}, \widehat{\boldsymbol{w}}^{T}\right]^{T}, and λ ^ λ ^ widehat(lambda)\widehat{\boldsymbol{\lambda}} and w ^ w ^ widehat(w)\widehat{\boldsymbol{w}} are vectors of all the parameters and weights associated with each symmetry group. The Levenberg-Marquardt (LM) algorithm [27, 28] is used to solve (15). The LM algorithm computes the step direction, h h h\boldsymbol{h} as
其中 τ = [ λ ^ T , w ^ T ] T τ = λ ^ T , w ^ T T tau=[ widehat(lambda)^(T), widehat(w)^(T)]^(T)\boldsymbol{\tau}=\left[\widehat{\boldsymbol{\lambda}}^{T}, \widehat{\boldsymbol{w}}^{T}\right]^{T} 和 λ ^ λ ^ widehat(lambda)\widehat{\boldsymbol{\lambda}} 以及 w ^ w ^ widehat(w)\widehat{\boldsymbol{w}} 是与每个对称组相关的所有参数和权重向量。Levenberg-Marquardt (LM) 算法 [27, 28] 用于求解 (15)。LM 算法计算步进方向 h h h\boldsymbol{h} 如下
h = − A + ȷ T g h = − A + ȷ T g h=-A^(+)ȷ^(T)g\boldsymbol{h}=-\mathrm{A}^{+} \boldsymbol{\jmath}^{T} \boldsymbol{g}
where A = J T J + ν diag ( J T J ) , ν > 0 A = J T J + ν diag J T J , ν > 0 A=J^(T)J+nu diag(J^(T)(J)),nu > 0\mathrm{A}=\mathrm{J}^{T} \mathrm{~J}+\nu \operatorname{diag}\left(\mathrm{J}^{T} \mathrm{~J}\right), \nu>0 is a parameter that controls the scale of exploration, and ( ⋅ ) + ( ⋅ ) + (*)^(+)(\cdot)^{+}denotes the Moore-Penrose pseudo-inverse. The matrix J ∈ R n b × n τ J ∈ R n b × n τ J inR^(n_(b)xxn_(tau))J \in \mathbb{R}^{n_{b} \times n_{\tau}} is the Jacobian matrix given by
其中， A = J T J + ν diag ( J T J ) , ν > 0 A = J T J + ν diag J T J , ν > 0 A=J^(T)J+nu diag(J^(T)(J)),nu > 0\mathrm{A}=\mathrm{J}^{T} \mathrm{~J}+\nu \operatorname{diag}\left(\mathrm{J}^{T} \mathrm{~J}\right), \nu>0 是控制探索规模的参数， ( ⋅ ) + ( ⋅ ) + (*)^(+)(\cdot)^{+} 表示摩尔-彭罗斯伪逆。矩阵 J ∈ R n b × n τ J ∈ R n b × n τ J inR^(n_(b)xxn_(tau))J \in \mathbb{R}^{n_{b} \times n_{\tau}} 是雅各布矩阵，其值为
J ( i , j ) = ∂ g i ∂ τ j J ( i , j ) = ∂ g i ∂ τ j J_((i,j))=(delg_(i))/(deltau_(j))\mathrm{J}_{(i, j)}=\frac{\partial \boldsymbol{g}_{i}}{\partial \boldsymbol{\tau}_{j}}
and n τ n τ n_(tau)n_{\tau} is the sum of the number of parameters and weights. The Jacobian can also be
和 n τ n τ n_(tau)n_{\tau} 是参数和权重的总和。Jacobian 也可以是
written in terms of block matrices as
用块矩阵写成
J = [ ∑ k = 1 d V x k T diag ( w ) ∂ y ( : , k ) ∂ λ ^ , V T ∂ w ∂ w ^ ] J = ∑ k = 1 d   V x k T diag ( w ) ∂ y ( : , k ) ∂ λ ^ , V T ∂ w ∂ w ^ J=[sum_(k=1)^(d)V_(x_(k))^(T)diag(w)(dely_((:,k)))/(del( widehat(lambda))),V^(T)(del w)/(del( widehat(w)))]\mathbf{J}=\left[\sum_{k=1}^{d} \mathbf{V}_{x_{k}}^{T} \operatorname{diag}(\boldsymbol{w}) \frac{\partial \boldsymbol{y}_{(:, k)}}{\partial \widehat{\boldsymbol{\lambda}}}, \mathbf{V}^{T} \frac{\partial \boldsymbol{w}}{\partial \widehat{\boldsymbol{w}}}\right]
where V x k V x k V_(x_(k))\mathrm{V}_{x_{k}} is the k k kk-th direction derivative of V and y ( : , k ) y ( : , k ) y_((:,k))\boldsymbol{y}_{(:, k)} is the k k kk-th direction component vector of y y y\boldsymbol{y}. The matrix ∂ y ( : , k ) / ∂ λ ^ ∂ y ( : , k ) / ∂ λ ^ dely_((:,k))//del widehat(lambda)\partial \boldsymbol{y}_{(:, k)} / \partial \widehat{\boldsymbol{\lambda}} is computed using the relation in (14), and ∂ w / ∂ w ^ ∂ w / ∂ w ^ del w//del widehat(w)\partial \boldsymbol{w} / \partial \widehat{\boldsymbol{w}} is a matrix of zeros and ones.
其中 V x k V x k V_(x_(k))\mathrm{V}_{x_{k}} 是 V 的 k k kk -th 方向导数， y ( : , k ) y ( : , k ) y_((:,k))\boldsymbol{y}_{(:, k)} 是 y y y\boldsymbol{y} 的 k k kk -th 方向分量向量。矩阵 ∂ y ( : , k ) / ∂ λ ^ ∂ y ( : , k ) / ∂ λ ^ dely_((:,k))//del widehat(lambda)\partial \boldsymbol{y}_{(:, k)} / \partial \widehat{\boldsymbol{\lambda}} 使用 (14) 中的关系式计算， ∂ w / ∂ w ^ ∂ w / ∂ w ^ del w//del widehat(w)\partial \boldsymbol{w} / \partial \widehat{\boldsymbol{w}} 是一个 0 和 1 的矩阵。
The algorithm starts with an initial guess, τ ( 0 ) τ ( 0 ) tau^((0))\boldsymbol{\tau}^{(0)}, and the value of τ τ tau\boldsymbol{\tau} at the n n nn-th iteration is updated as
算法从初始猜测 τ ( 0 ) τ ( 0 ) tau^((0))\boldsymbol{\tau}^{(0)} 开始，第 n n nn 次迭代时 τ τ tau\boldsymbol{\tau} 的值更新为
τ ( n + 1 ) = τ ( n ) + η ( n ) h ( n ) , τ ( n + 1 ) = τ ( n ) + η ( n ) h ( n ) , tau^((n+1))=tau^((n))+eta^((n))h^((n)),\boldsymbol{\tau}^{(n+1)}=\boldsymbol{\tau}^{(n)}+\eta^{(n)} \boldsymbol{h}^{(n)},
where η ( n ) = 1 η ( n ) = 1 eta^((n))=1\eta^{(n)}=1 is used unless a negative weight is encountered. If a negative weight is encountered at the i i ii-th entry of τ ( n + 1 ) τ ( n + 1 ) tau^((n+1))\boldsymbol{\tau}^{(n+1)}, then the update is recomputed using
其中 η ( n ) = 1 η ( n ) = 1 eta^((n))=1\eta^{(n)}=1 将被使用，除非遇到负权重。如果在 τ ( n + 1 ) τ ( n + 1 ) tau^((n+1))\boldsymbol{\tau}^{(n+1)} 的 i i ii -th 条目中遇到负权重，那么更新将使用
η ( n ) = ( ε − τ i ( n ) ) h i ( n ) η ( n ) = ε − τ i ( n ) h i ( n ) eta^((n))=((epsi-tau_(i)^((n))))/(h_(i)^((n)))\eta^{(n)}=\frac{\left(\varepsilon-\boldsymbol{\tau}_{i}^{(n)}\right)}{\boldsymbol{h}_{i}^{(n)}}
where ε > 0 ε > 0 epsi > 0\varepsilon>0 is an arbitrary lower bound for the update of the negative weight, e.g., we use ε = 10 − 4 ε = 10 − 4 epsi=10^(-4)\varepsilon=10^{-4}. All of the quadrature rules in this work are obtained using the open-source Julia code SummationByParts.jl [29].
其中 ε > 0 ε > 0 epsi > 0\varepsilon>0 是负权重更新的任意下限，例如，我们使用 ε = 10 − 4 ε = 10 − 4 epsi=10^(-4)\varepsilon=10^{-4} 。本研究中的所有正交规则都是通过开源的 Julia 代码 SummationByParts.jl [29] 获得的。

4. Initial Guess Generation
4.生成初始猜测

The initial guesses for the PI quadrature rules are generated by forming tensor-product structures on quadrilateral/hexahedral subdomains of the simplices using the LG nodes lying in the first half of the line reference element. The construction procedure is best explained through an example. Consider the construction of the degree q = 8 q = 8 q=8q=8 line-LG quadrature rule on the triangle; we first map the nodes of the degree q = 9 q = 9 q=9q=9 one-dimensional LG quadrature rule that lie in the first half of the line reference element onto the line connecting the bottom left vertex to the bottom edge midpoint. We apply the same mapping to the line connecting the left edge midpoint to the centroid of the triangle. Then we connect the corresponding LG nodes on the two lines and, once again, apply the mapping to each of the connecting lines to find the initial nodes on a quadrilateral subdomain of the triangle, see Fig. 2a. The unique symmetry orbits can be found using the coordinates of the nodes that lie in one of the triangles obtained by dividing one of the three quadrilateral subdomains into two triangles, as shown in Fig. 2b. Denoting the number of nodes in the unique orbits and their nodal coordinate matrix by n q ′ n q ′ n_(q)^(')n_{q}^{\prime} and y ′ ∈ R d × n q ′ y ′ ∈ R d × n q ′ y^(')inR^(d xxn_(q)^('))\boldsymbol{y}^{\prime} \in \mathbb{R}^{d \times n_{q}^{\prime}}, respectively, we can compute the barycentric coordinates using the relation in (14) and the fact that the sum of the barycentric coordinates of each node is unity, i.e.,
通过使用位于线参考元素前半部分的 LG 节点，在四边形/六面体子域上形成张量乘积结构，从而生成 PI 正交规则的初始猜测。通过一个例子可以很好地解释构造过程。考虑在三角形上构建度数为 q = 8 q = 8 q=8q=8 的线 LG 正交规则；我们首先将位于线参考元素前半部分的度数为 q = 9 q = 9 q=9q=9 的一维 LG 正交规则的节点映射到连接左下顶点和底边中点的直线上。我们将同样的映射应用到连接左边缘中点和三角形中心点的直线上。然后，我们将两条线上相应的 LG 节点连接起来，并再次对每条连接线应用映射，以找到三角形四边形子域上的初始节点，见图 2a。如图 2b 所示，使用位于将三个四边形子域中的一个划分为两个三角形后得到的其中一个三角形中的节点坐标，可以找到唯一的对称轨道。用 n q ′ n q ′ n_(q)^(')n_{q}^{\prime} 和 y ′ ∈ R d × n q ′ y ′ ∈ R d × n q ′ y^(')inR^(d xxn_(q)^('))\boldsymbol{y}^{\prime} \in \mathbb{R}^{d \times n_{q}^{\prime}} 分别表示独特轨道中的节点数及其节点坐标矩阵，我们可以利用 (14) 中的关系和每个节点的偏心坐标之和为一的事实计算偏心坐标，即
λ = ( T ― T ) − 1 y ― ′ , λ = T _ T − 1 y _ ′ , lambda=(T_^(T))^(-1)y_^('),\boldsymbol{\lambda}=\left(\underline{\mathrm{T}}^{T}\right)^{-1} \underline{\boldsymbol{y}}^{\prime},
where ( ( ( (( ) denotes concatenation of an additional row vector of ones at the bottom of a matrix. The complete nodal distribution on the triangle shown in Fig. 2c can be computed using the barycentric coordinates and the symmetry relations in the first line of (3). For the tetrahedron, the unique symmetry orbits are provided by the nodes that lie in one of the
其中 ( ( ( (( ) 表示矩阵底部的额外 1 行向量的连接。图 2c 所示三角形上完整的节点分布可以通过重心坐标和 (3) 第一行中的对称关系计算出来。对于四面体来说，唯一的对称轨道由位于其中一个对称轨道上的节点提供。
Figure 2: Generation of initial nodes for the degree q = 4 q = 4 q=4q=4 line-LG quadrature rule on triangles and tetrahedra.
图 2：为三角形和正四面体上的 q = 4 q = 4 q=4q=4 线-LG 正交规则生成初始节点。
six tetrahedra obtained by dividing one of the four hexahedral subdomains of the reference tetrahedron into six tetrahedra, see Fig. 2e.
将参照四面体的四个六面体子域中的一个分成六个四面体后得到的六个四面体，见图 2e。
The initial weight parameters associated with the unique symmetry orbits are set to a constant value of ( 2 / q ) 3 ( 2 / q ) 3 (2//q)^(3)(2 / q)^{3} in all cases, as the solution to the line-LG quadrature problem is observed to be relatively less sensitive to the values of the weights than the location of the nodes.
在所有情况下，与唯一对称轨道相关的初始权重参数都被设置为 ( 2 / q ) 3 ( 2 / q ) 3 (2//q)^(3)(2 / q)^{3} 的恒定值，因为根据观察，线-LG 正交问题的解对权重值的敏感度相对低于对节点位置的敏感度。

5. The Line-LG Quadrature Rule
5.线性-LG 正交规则

With the node parameters and weights set for each symmetry group, we proceed to solve the minimization problem in (15) using the LM algorithm. We derived line-LG quadrature rules up to degree q = 84 q = 84 q=84q=84 and q = 40 q = 40 q=40q=40 on the triangle and tetrahedron, respectively. The reason we stopped at q = 84 q = 84 q=84q=84 for the triangle is because the outputs of the classic Gamma function used in the construction of the PKD basis functions are out of the range of numbers that can be represented in double precision. For the tetrahedron, memory limitations become significant at quadrature degrees slightly larger than forty. While both of these limitations can be circumvented, we chose not to pursue this task and have left it for future work.
在为每个对称组设置了节点参数和权重后，我们开始使用 LM 算法求解 (15) 中的最小化问题。我们在三角形和四面体上分别推导出了阶数最高为 q = 84 q = 84 q=84q=84 和 q = 40 q = 40 q=40q=40 的线性-线性方程组正交规则。对于三角形，我们之所以在 q = 84 q = 84 q=84q=84 时停止，是因为在构建 PKD 基函数时使用的经典 Gamma 函数的输出超出了可以用双精度表示的数字范围。对于四面体来说，当正交度略大于 40 时，内存限制就变得非常明显。虽然这两个限制都可以规避，但我们选择不进行这项工作，而是将其留给未来的工作。
The type and number of symmetry orbits in the line-LG quadrature rules can be determined by studying the structure of the initial nodal distribution. The one-dimensional LG quadrature rule is exact for polynomials up to degree q = 2 n 1 − 1 q = 2 n 1 − 1 q=2n_(1)-1q=2 n_{1}-1, where n 1 n 1 n_(1)n_{1} is the number of quadrature points on the reference line. Defining m = ( n 1 mod 2 ) m = n 1 mod 2 m=(n_(1)mod2)m=\left(n_{1} \bmod 2\right) and n r = ( n 1 − m ) / 2 n r = n 1 − m / 2 n_(r)=(n_(1)-m)//2n_{r}=\left(n_{1}-m\right) / 2, the number of orbits in each type of symmetry group of the triangle can be calculated as
通过研究初始结点分布的结构，可以确定线-LG 正交规则中对称轨道的类型和数量。一维 LG 正交规则对于阶数不超过 q = 2 n 1 − 1 q = 2 n 1 − 1 q=2n_(1)-1q=2 n_{1}-1 的多项式是精确的，其中 n 1 n 1 n_(1)n_{1} 是参考线上的正交点数。根据 m = ( n 1 mod 2 ) m = n 1 mod 2 m=(n_(1)mod2)m=\left(n_{1} \bmod 2\right) 和 n r = ( n 1 − m ) / 2 n r = n 1 − m / 2 n_(r)=(n_(1)-m)//2n_{r}=\left(n_{1}-m\right) / 2 的定义，可以计算出三角形各对称组中的轨道数为
| S 1 | = m , | S 21 | = ( 1 + m ) n r , | S 111 | = ( n r 2 − n r ) / 2 S 1 = m , S 21 = ( 1 + m ) n r , S 111 = n r 2 − n r / 2 |S_(1)|=m,quad|S_(21)|=(1+m)n_(r),quad|S_(111)|=(n_(r)^(2)-n_(r))//2\left|S_{1}\right|=m, \quad\left|S_{21}\right|=(1+m) n_{r}, \quad\left|S_{111}\right|=\left(n_{r}^{2}-n_{r}\right) / 2
Similarly, the type and numbers of symmetry orbits on the reference tetrahedron are
同样，参照四面体上对称轨道的类型和数量为
| S 1 | = m , | S 31 | = ( 1 + m ) n r , | S 22 | = m n r , | S 211 | = 1 + 2 m 1 + m ( n r 2 − n r ) , | S 1111 | = ( n r − 1 ) 3 − n r + 1 6 . S 1 = m , S 31 = ( 1 + m ) n r , S 22 = m n r , S 211 = 1 + 2 m 1 + m n r 2 − n r , S 1111 = n r − 1 3 − n r + 1 6 . {:[|S_(1)|=m","quad|S_(31)|=(1+m)n_(r)","quad|S_(22)|=mn_(r)","],[|S_(211)|=(1+2m)/(1+m)(n_(r)^(2)-n_(r))","quad|S_(1111)|=((n_(r)-1)^(3)-n_(r)+1)/(6).]:}\begin{aligned} \left|S_{1}\right| & =m, \quad\left|S_{31}\right|=(1+m) n_{r}, \quad\left|S_{22}\right|=m n_{r}, \\ \left|S_{211}\right| & =\frac{1+2 m}{1+m}\left(n_{r}^{2}-n_{r}\right), \quad\left|S_{1111}\right|=\frac{\left(n_{r}-1\right)^{3}-n_{r}+1}{6} . \end{aligned}
Note that the above numbers of orbits are functions of n 1 n 1 n_(1)n_{1} only; hence, they provide information for the initial guesses generated using LG quadrature rules with n 1 n 1 n_(1)n_{1} nodes. The actual derivation of a degree q q qq line-LG quadrature rule may use n 1 n 1 n_(1)n_{1} values that correspond to a higher degree one-dimensional LG quadrature rule, as explained in the next section.
请注意，上述轨道数仅是 n 1 n 1 n_(1)n_{1} 的函数；因此，它们为使用具有 n 1 n 1 n_(1)n_{1} 节点的 LG 正交规则生成的初始猜测提供了信息。在实际推导阶数为 q q qq 的线 LG 正交规则时，可能会使用 n 1 n 1 n_(1)n_{1} 值，这些值与更高阶数的一维 LG 正交规则相对应，这将在下一节中解释。

5.1. Efficiency of the Line-LG Quadrature Rule
5.1.线性-LG 正交规则的效率

The efficiency of the line-LG quadrature rules depends on the number of one-dimensional quadrature points, n 1 n 1 n_(1)n_{1}, used in their construction and the polynomial degree they can integrate exactly. On the triangle, we have observed some quadrature accuracy patterns associated with the degree and values of n 1 n 1 n_(1)n_{1}. If q − 1 q − 1 q-1q-1 is divisible by 4 , then only n 1 = ⌊ q / 2 ⌋ + 1 n 1 = ⌊ q / 2 ⌋ + 1 n_(1)=|__ q//2__|+1n_{1}=\lfloor q / 2\rfloor+1 nodes are needed to derive the line-LG quadrature rule. This case coincides with the presence of a mid-point in the one-dimensional LG quadrature rule. If q q qq is even or q ≥ 30 q ≥ 30 q >= 30q \geq 30, then n 1 = ⌊ q / 2 ⌋ + 1 n 1 = ⌊ q / 2 ⌋ + 1 n_(1)=|__ q//2__|+1n_{1}=\lfloor q / 2\rfloor+1 nodes are required to derive the line-LG quadrature rules. Finally, if q < 30 q < 30 q < 30q<30, odd, and q − 1 q − 1 q-1q-1 is not divisible by 4 , then n 1 = ⌊ q / 2 ⌋ + 2 n 1 = ⌊ q / 2 ⌋ + 2 n_(1)=|__ q//2__|+2n_{1}=\lfloor q / 2\rfloor+2 nodes are required. For example, the degree 8 , 9 , 10 8 , 9 , 10 8,9,108,9,10, and 11 line-LG quadrature rules require initial guesses constructed with n 1 n 1 n_(1)n_{1} values of 5 , 5 , 6 5 , 5 , 6 5,5,65,5,6, and 7 , respectively. This has an impact on the efficiency of the rules, in the sense of definition (13), as, for example, the q = 9 q = 9 q=9q=9 rule has larger lower bounds on the number of nodes, but it has the same number of nodes as the q = 8 q = 8 q=8q=8 rule.
线性 LG 正交规则的效率取决于其构造中所使用的一维正交点 n 1 n 1 n_(1)n_{1} 的数量以及它们能够精确积分的多项式度数。在三角形上，我们观察到一些与 n 1 n 1 n_(1)n_{1} 的度数和值相关的正交精度模式。如果 q − 1 q − 1 q-1q-1 能被 4 整除，那么只需要 n 1 = ⌊ q / 2 ⌋ + 1 n 1 = ⌊ q / 2 ⌋ + 1 n_(1)=|__ q//2__|+1n_{1}=\lfloor q / 2\rfloor+1 节点就能推导出线性 LG 正交规则。这种情况与一维 LG 正交规则中存在中点的情况相吻合。如果 q q qq 是偶数或 q ≥ 30 q ≥ 30 q >= 30q \geq 30 ，则需要 n 1 = ⌊ q / 2 ⌋ + 1 n 1 = ⌊ q / 2 ⌋ + 1 n_(1)=|__ q//2__|+1n_{1}=\lfloor q / 2\rfloor+1 节点来推导线-LG 正交规则。最后，如果 q < 30 q < 30 q < 30q<30 为奇数，且 q − 1 q − 1 q-1q-1 不能被 4 整除，则需要 n 1 = ⌊ q / 2 ⌋ + 2 n 1 = ⌊ q / 2 ⌋ + 2 n_(1)=|__ q//2__|+2n_{1}=\lfloor q / 2\rfloor+2 节点。例如，阶数 8 , 9 , 10 8 , 9 , 10 8,9,108,9,10 和 11 线-LG 正交规则需要分别以 5 , 5 , 6 5 , 5 , 6 5,5,65,5,6 和 7 的 n 1 n 1 n_(1)n_{1} 值构建初始猜测。例如， q = 9 q = 9 q=9q=9 规则的节点数下限更大，但其节点数与 q = 8 q = 8 q=8q=8 规则相同。
On the tetrahedron, a more consistent pattern is observed. For all values of q q qq except 3, 7, and 11 , we were able to find line-LG quadrature rule using n 1 = ⌊ q / 2 ⌋ + 1 n 1 = ⌊ q / 2 ⌋ + 1 n_(1)=|__ q//2__|+1n_{1}=\lfloor q / 2\rfloor+1 nodes to construct the initial guesses. For the exceptions, n 1 = ⌊ q / 2 ⌋ + 2 n 1 = ⌊ q / 2 ⌋ + 2 n_(1)=|__ q//2__|+2n_{1}=\lfloor q / 2\rfloor+2 nodes are required. The solutions for the line-LG quadrature rules are obtained in a few iterations of the LM algorithm. For example, the q = 20 q = 20 q=20q=20 line-LG rule on the tetrahedron requires less than 10 iterations.
在四面体上，我们观察到一种更为一致的模式。对于除 3、7 和 11 以外的所有 q q qq 值，我们都能使用 n 1 = ⌊ q / 2 ⌋ + 1 n 1 = ⌊ q / 2 ⌋ + 1 n_(1)=|__ q//2__|+1n_{1}=\lfloor q / 2\rfloor+1 节点构建初始猜测，从而找到线性 LG 正交规则。对于例外情况，需要使用 n 1 = ⌊ q / 2 ⌋ + 2 n 1 = ⌊ q / 2 ⌋ + 2 n_(1)=|__ q//2__|+2n_{1}=\lfloor q / 2\rfloor+2 节点。线性-LG 正交规则的解可以通过 LM 算法的几次迭代获得。例如，四面体上的 q = 20 q = 20 q=20q=20 线-LG 规则只需不到 10 次迭代。
Despite their tensor-product-like initial guess structures and ease of derivation, the lineLG quadrature rules are surprisingly efficient. Fig. 3 shows their efficiency relative to other known quadrature rules in the literature. For triangles the line-LG rules are above 80 % 80 % 80%80 \% efficient for most degrees, and are close to 90 % 90 % 90%90 \% efficient at very high degrees. For the tetrahedron, they are more than 60 % 60 % 60%60 \% efficient for most degrees, which is significant given that many of the existing rules in the literature, except the recent additions of Chuluunbaatar et al. [6], are less than 80 % 80 % 80%80 \% efficient for degrees greater than ten.
尽管线性 LG 正交规则的初始猜测结构类似于张量乘积，而且易于推导，但其效率却出人意料。图 3 显示了相对于文献中其他已知正交规则的效率。对于三角形，线性-线性正交规则在大多数度数下的效率高于 80 % 80 % 80%80 \% ，在极高的度数下效率接近 90 % 90 % 90%90 \% 。对于四面体，它们在大多数度数下的效率都高于 60 % 60 % 60%60 \% ，这一点非常重要，因为除了 Chuluunbaatar 等人 [6] 最近增加的规则外，文献中许多现有规则在大于 10 度时的效率都低于 80 % 80 % 80%80 \% 。

6. Node Elimination 6.节点消除

Node elimination is a technique used to create quadrature rules with fewer nodes, starting from existing known rules [3]. The idea is straightforward; given a quadrature rule, eliminate an orbit and attempt to solve the minimization problem in (15) to machine precision. If the attempt is unsuccessful, reintroduce the eliminated orbit. If successful, a new quadrature rule is obtained, and the process is repeated until no further orbit can be eliminated. When there are several orbits that can be eliminated at a given stage, different criteria can be used to
节点消除是一种用于从现有已知规则出发，创建节点更少的正交规则的技术[3]。其原理简单明了：给定一个正交规则，消除一个轨道，并尝试以机器精度求解 (15) 中的最小化问题。如果尝试不成功，则重新引入被消除的轨道。如果成功，则得到一个新的正交规则，然后重复这一过程，直到无法再消除任何轨道为止。当某一阶段有多个轨道可以消除时，可以使用不同的标准来消除它们
Figure 3: Efficiency of the line-LG quadrature rules relative to existing symmetric quadrature rules on triangles (a) and tetrahedra (b).
图 3：相对于三角形（a）和四面体（b）上现有的对称正交规则，线-线性正交规则的效率。
decide which orbit to eliminate. For example, Xiao and Gimbutas [3] proposed a significance index for each node, which is the sum or weighted sum of the squares of the basis functions evaluated at each node. The node with the smallest significance index is considered to have the lowest contribution to the integral approximation; hence, it is eliminated first. Another approach is to eliminate the node with the smallest weight, which is appealing from the perspective of collocated high-order methods since this will reduce the norm of the inverse of the mass or norm matrix and, consequently, the conditioning of the system matrices resulting from discretizations of PDEs. One can also eliminate orbits such that the minimum distance between nodes is increased, which can improve the CFL number for numerical solutions of PDEs using explicit time-marching methods. It may also be desirable to eliminate nodes that are very close to the element boundaries. While all of these options appear reasonable, it is not clear a priori which approach will ultimately result in the minimum number of nodes. The decision at each stage will affect the set of orbits that can be eliminated in subsequent stages, thereby determining the total number of nodes that can be attained. Due to the combinatorial nature of the elimination process, it is impractical to test all possible paths, particularly at higher degrees. In this work, we have not made attempts to investigate the optimal node elimination strategy. However, based on limited experiments, we have opted to eliminate orbits that produce nodes closest to the facets or that have the smallest weights. For triangles, we have used both of these approaches and neither is conclusively better than the other. For the tetrahedron, the first approach is used in most cases, but when efficiency of a derived quadrature is low, both approaches are applied to check if improvements can be made.
决定消除哪些轨道。例如，Xiao 和 Gimbutas [3] 为每个节点提出了一个重要指数，即在每个节点评估的基函数平方的总和或加权和。重要指数最小的节点被认为对积分近似的贡献最小，因此首先被剔除。另一种方法是剔除权重最小的节点，从同位高阶方法的角度来看，这种方法很有吸引力，因为这将减少质量或规范矩阵逆矩阵的规范，从而减少 PDE 离散化产生的系统矩阵的调节。我们还可以消除轨道，从而增加节点之间的最小距离，这可以提高使用显式时间进动方法对 PDE 进行数值求解的 CFL 数。此外，还可以消除非常靠近元素边界的节点。虽然所有这些方案看起来都很合理，但并不清楚哪种方法最终能使节点数量最少。每个阶段的决定都会影响到后续阶段可以消除的轨道集，从而决定可以达到的节点总数。由于消除过程的组合性质，测试所有可能的路径是不切实际的，尤其是在较高的度数下。在这项工作中，我们没有尝试研究最优节点消除策略。不过，根据有限的实验，我们选择了消除产生最接近切面节点或权重最小节点的轨道。对于三角形，我们同时使用了这两种方法，但两者都没有明显的优越性。 对于四面体，大多数情况下采用第一种方法，但当推导的正交效率较低时，则同时采用两种方法，以检查是否可以改进。
The node elimination process takes into account the structure of the lower-bound estimates presented in (7) and (11). In particular, for the first two outer elimination iterations, see Fig. 1, the number of nodes in each symmetry orbit is kept equal to or greater than the number of nodes presented in the estimates. This increases the possibility of attaining
节点消除过程考虑了 (7) 和 (11) 中给出的下限估计值的结构。特别是在前两次外部消除迭代中（见图 1），每个对称轨道中的节点数保持等于或大于估计值中的节点数。这增加了获得
Table 1: Number of nodes of the newly derived symmetric PI quadrature rules on triangles compared to existing rules. New quadrature rules with fewer nodes than existing rules and those for higher degrees than previously available are underlined.
表 1：与现有规则相比，新推导的三角形对称 PI 正交规则的节点数。下划线表示节点数少于现有规则的新正交规则，以及度数高于现有规则的新正交规则。

quadrature rules that match the estimated bounds or have a few more orbits than the estimates. Furthermore, at each stage in the first two outer iterations, the type of orbit that supports the highest number of nodes is eliminated until no further orbit of that kind can be eliminated. For example, at any stage in the first two outer iterations, the S 111 S 111 S_(111)S_{111} orbits are ranked based on proximity to the facets or based on their weights, and the highest ranking orbit is eliminated before any of the S 21 S 21 S_(21)S_{21} orbits are considered. After the second outer iteration, all types of symmetry orbits are ranked at once and the highest ranking orbit is eliminated. The node elimination process is sensitive to the value of ν ν nu\nu used in the LM algorithm, which is defined in (16). Hence, values of ν ν nu\nu in orders of 10 ranging from 10 − 8 10 − 8 10^(-8)10^{-8} to 10 4 10 4 10^(4)10^{4} are used. Admittedly, the process of node elimination requires changing the above parameters manually, starting from higher degree line-LG rules, or, in rare cases, combining two or more line-LG quadrature rules to improve efficiency. Automation and robust elimination algorithms are lacking and are worth pursuing in order to further improve efficiency and extend the proposed approach to dimensions greater than three.
在前两个外部迭代阶段的每个阶段，都会剔除支持节点数最多的轨道类型，直到没有其他轨道可以被剔除为止。此外，在前两次外部迭代的每个阶段，都会淘汰支持节点数最多的轨道类型，直到无法再淘汰该类轨道为止。例如，在前两次外部迭代的任何阶段， S 111 S 111 S_(111)S_{111} 轨道都会根据与切面的接近程度或权重进行排序，在考虑任何 S 21 S 21 S_(21)S_{21} 轨道之前，都会先淘汰排序最高的轨道。在第二次外部迭代之后，所有类型的对称轨道都会被一次性排序，排序最高的轨道会被淘汰。节点消除过程对 LM 算法中使用的 ν ν nu\nu 值很敏感，该值定义在 (16) 中。因此，使用的 ν ν nu\nu 值的数量级为 10，范围从 10 − 8 10 − 8 10^(-8)10^{-8} 到 10 4 10 4 10^(4)10^{4} 。诚然，节点消除过程需要手动更改上述参数，从更高阶的线性 LG 规则开始，或者在极少数情况下，结合两个或更多的线性 LG 正交规则来提高效率。目前还缺乏自动化和稳健的消除算法，为了进一步提高效率，并将所提出的方法扩展到三维以上维度，这些算法值得研究。

7. New Symmetric PI Quadrature Rules
7.新的对称 PI 正交规则

By applying the strategies outlined in the previous sections, we derived more efficient quadrature rules than the line-LG rules on both triangles and tetrahedra. The number of nodes for the quadrature rules on triangles are shown in Table 1. While a few of the newly derived rules on the triangle have more nodes than the existing rules, most of them have either an equal number or fewer nodes. Table 2 shows that many of the derived rules on the tetrahedron have fewer nodes than those presented by Jaśkowiec and Sukumar [5], but all of the rules up to degree twenty have an equal number or more nodes than the recent results of Chuluunbaatar et al. [6]. To the authors’ knowledge all of the rules greater than degree twenty are new. The nodal locations of some of the derived quadrature rules are shown in Fig. 4.
通过应用前几节概述的策略，我们在三角形和正四面体上都得出了比线性 LG 规则更有效的正交规则。三角形正交规则的节点数如表 1 所示。虽然在三角形上新推导出的规则中，有少数规则的节点数多于现有规则，但大多数规则的节点数要么相同，要么更少。表 2 显示，许多推导出的正四面体规则的节点数少于 Jaśkowiec 和 Sukumar [5]提出的规则，但 20 度以内的所有规则的节点数都等于或多于 Chuluunbaatar 等人 [6] 的最新结果。据作者所知，所有大于 20 度的规则都是新规则。图 4 显示了一些衍生正交规则的节点位置。
The efficiency of the quadrature rules obtained after applying node elimination is shown in Fig. 5 for both the triangle and tetrahedron. Most of the quadrature rules for the triangle are more than 95 % 95 % 95%95 \% efficient, while all but a few are more than 80 % 80 % 80%80 \% efficient for the tetrahedron.
图 5 显示了应用节点消除后得到的三角形和四面体正交规则的效率。对于三角形，大多数正交规则的效率都高于 95 % 95 % 95%95 \% ，而对于四面体，除了少数规则外，所有规则的效率都高于 80 % 80 % 80%80 \% 。
Table 2: Number of nodes of the newly derived symmetric PI quadrature rules on tetrahedra compared to existing rules. New quadrature rules for higher degrees than previously available are underlined.
表 2：与现有规则相比，新推导的四面体对称 PI 正交规则的节点数。下划线部分为新的正交规则，适用于比现有规则更高的度数。

Figure 4: Examples of symmetric PI quadrature rules on triangles and tetrahedra.
图 4：三角形和正四面体的对称 PI 正交规则示例。
It is evident from Fig. 5b that most of the newly derived quadrature rules on the tetrahedron have comparable efficiency relative to the most efficient existing rules, with at most a 7 % 7 % 7%7 \% reduction in efficiency.
从图 5b 中可以明显看出，大多数新推导出的四面体正交规则与现有最高效规则的效率相当，最多只是降低了 7 % 7 % 7%7 \% 。
All of the quadrature rules are provided in the supplementary material accompanying this paper. The line-LG quadrature rules are also included, which could play an important role in future developments of robust node elimination algorithms and derivation of more efficient quadrature rules.
本文所附的补充材料中提供了所有的正交规则。其中还包括线-LG 正交规则，这对今后开发鲁棒节点消除算法和推导更高效的正交规则具有重要作用。

8. Numerical Results 8.数值结果

We apply the derived quadrature rules to compute the following integrals of highly oscillatory functions,
我们运用推导出的正交规则计算下列高度振荡函数的积分、
I 2 = ∫ 0 1 ∫ 0 1 sin ⁡ ( 48 π x 1 8 ) cos ⁡ ( 48 π x 2 5 ) d x d y I 3 = ∫ 0 1 ∫ 0 1 ∫ 0 1 sin ⁡ ( 16 π x 1 8 ) cos ⁡ ( 16 π x 2 5 ) cos ⁡ ( 16 π x 3 7 ) d x d y d z I 2 = ∫ 0 1   ∫ 0 1   sin ⁡ 48 π x 1 8 cos ⁡ 48 π x 2 5 d x d y I 3 = ∫ 0 1   ∫ 0 1   ∫ 0 1   sin ⁡ 16 π x 1 8 cos ⁡ 16 π x 2 5 cos ⁡ 16 π x 3 7 d x d y d z {:[I_(2)=int_(0)^(1)int_(0)^(1)sin(48 pix_(1)^(8))cos(48 pix_(2)^(5))dxdy],[I_(3)=int_(0)^(1)int_(0)^(1)int_(0)^(1)sin(16 pix_(1)^(8))cos(16 pix_(2)^(5))cos(16 pix_(3)^(7))dxdydz]:}\begin{aligned} & \mathcal{I}_{2}=\int_{0}^{1} \int_{0}^{1} \sin \left(48 \pi x_{1}^{8}\right) \cos \left(48 \pi x_{2}^{5}\right) \mathrm{d} x \mathrm{~d} y \\ & \mathcal{I}_{3}=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \sin \left(16 \pi x_{1}^{8}\right) \cos \left(16 \pi x_{2}^{5}\right) \cos \left(16 \pi x_{3}^{7}\right) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z \end{aligned}
Figure 5: Efficiency of the quadrature rules derived in this work relative to existing symmetric quadrature rules on triangles (a) and tetrahedra (b).
图 5：相对于现有的三角形（a）和正四面体（b）对称正交规则，本研究得出的正交规则的效率。
in two and three dimensions, respectively. The exact values of these integrals can be obtained using symbolic mathematics libraries, and the first sixteen significant digits are 0.03116210698718051 and 0.02288392144769807 , respectively. To test the accuracy of the quadrature rules, we constructed meshes for the unit square and unit cube that maintained an approximately equal number of degrees of freedom across different quadrature degrees. For the square, the meshes were constructed such that the total number of nodes was within 7 % 7 % 7%7 \% of 567450 . For the cube, the tetrahedral meshes were constructed such that the total number of nodes was within 7 % 7 % 7%7 \% of 158779350 . The stated reference total number of nodes correspond to having 450 degree q = 84 q = 84 q=84q=84 triangular elements and 41,154 degree q = 40 q = 40 q=40q=40 tetrahedral elements with the new quadrature rules (we remind the reader that the label “new” refers to the quadrature rules obtained after node elimination).
分别为二维和三维。这些积分的精确值可通过符号数学库获得，前十六位有效数字分别为 0.03116210698718051 和 0.02288392144769807。为了测试正交规则的准确性，我们为单位正方体和单位立方体构建了网格，使不同正交度的自由度数大致相等。对于正方体，网格的构建使节点总数在 567450 的 7 % 7 % 7%7 \% 范围内。对于立方体，四面体网格的构建使节点总数在 7 % 7 % 7%7 \% 158779350 的范围内。所述参考节点总数对应于采用新正交规则的 450 个阶数 q = 84 q = 84 q=84q=84 三角形元素和 41154 个阶数 q = 40 q = 40 q=40q=40 四面体元素（我们提醒读者，"新 "指的是消除节点后得到的正交规则）。
Fig. 6 shows the absolute values of the quadrature approximation errors of the integrals in (24) and (25). The new quadrature rules yield better approximations than the line-LG rules in most cases. Furthermore, they show comparable accuracy relative to the existing rules of [3] and [5] in all but some isolated cases. In particular, the new quadrature rules yield significantly less accurate results for the degree 10 case in three dimensions and degree 43 and 46 in two dimensions.
图 6 显示了 (24) 和 (25) 中积分的正交近似误差的绝对值。在大多数情况下，新的正交规则比线性-LG 规则产生了更好的近似值。此外，除个别情况外，它们在所有情况下都显示出与 [3] 和 [5] 现有规则相当的精度。特别是，新的正交规则在三维的 10 度情况和二维的 43 度和 46 度情况下产生的结果精度明显较低。
To investigate the three-dimensional case further, we apply the degree 8 through 12 quadrature rules to approximate the integral of a less oscillatory function,
为了进一步研究三维情况，我们应用 8 至 12 度正交规则来近似求一个振荡较小的函数的积分、
J 3 = ∫ 0 1 ∫ 0 1 ∫ 0 1 sin ⁡ ( 3 π x ) sin ⁡ ( 5 π y ) sin ⁡ ( 3 π z ) d x d y d z = 8 45 π 3 J 3 = ∫ 0 1   ∫ 0 1   ∫ 0 1   sin ⁡ ( 3 π x ) sin ⁡ ( 5 π y ) sin ⁡ ( 3 π z ) d x d y d z = 8 45 π 3 J_(3)=int_(0)^(1)int_(0)^(1)int_(0)^(1)sin(3pi x)sin(5pi y)sin(3pi z)dxdydz=(8)/(45pi^(3))\mathcal{J}_{3}=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \sin (3 \pi x) \sin (5 \pi y) \sin (3 \pi z) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z=\frac{8}{45 \pi^{3}}
Table 3 shows the quadrature errors due to approximation of the integral in (26) and the convergence rates obtained with various rules of degrees 8 through 12. It shows that the convergence rates for the even and odd degree quadrature rules are close to p + 2 p + 2 p+2p+2 and p + 1 p + 1 p+1p+1,
表 3 显示了 (26) 中积分近似引起的正交误差，以及使用 8 至 12 度各种规则得到的收敛率。结果表明，偶数度和奇数度正交规则的收敛率分别接近 p + 2 p + 2 p+2p+2 和 p + 1 p + 1 p+1p+1 、
Figure 6: Quadrature approximation absolute errors on triangles (a) and tetrahedra (b). The meshes used for each quadrature rule have approximately equal number of degrees of freedom.
图 6：三角形（a）和四面体（b）的正交近似绝对误差。每个正交规则使用的网格自由度大致相同。
respectively. These rates are in agreement with those reported in [6, 15] for quadrature rules on the tetrahedron and four-dimensional simplex. All the quadrature rules, except the degree 8 and 12 rules of [6], the degree 11 line-LG rule, and the new degree 12 rule, achieve comparable accuracy levels and convergence rates. As noted in Section 5.1, the q = 11 q = 11 q=11q=11 lineLG rule requires the same number of nodes as the q = 12 q = 12 q=12q=12 line-LG rules; hence, identical rules are used for both cases. The new degree 12 quadrature rule of this work and that of [6] seem to be significantly more accurate than the same degree quadrature rule in [5] and the q = 12 q = 12 q=12q=12 line-LG quadrature rule. They also achieve better accuracy for the highly oscillatory test case, as illustrated in Fig. 6b. On the other hand, despite the results in Fig. 6b, the q = 10 q = 10 q=10q=10 rule in [5] does not produce better results than the line-LG and the new quadrature rules when approximating (26), suggesting that the improvement observed when approximating (25) is specific to the problem or mesh considered. It is not immediately clear why some of the same degree quadrature rules yield significantly different accuracy levels. Investigating ways to improve the accuracy of quadrature rules and operators that are derived using them, e.g., SBP derivative operators, can lead to drastic improvements in efficiency of numerical methods for PDEs, and is left for future studies.
分别为这些收敛率与文献[6, 15]中报道的四面体和四维简体正交规则的收敛率一致。除了 [6] 中的 8 度和 12 度规则、11 度线-LG 规则和新的 12 度规则外，所有正交规则都达到了相当的精度水平和收敛速率。如第 5.1 节所述， q = 11 q = 11 q=11q=11 线性 LG 规则与 q = 12 q = 12 q=12q=12 线性 LG 规则所需的节点数相同；因此，两种情况下使用的规则完全相同。与 [5] 中的同度正交规则和 q = 12 q = 12 q=12q=12 line-LG 正交规则相比，本文和 [6] 中的新 12 度正交规则似乎更加精确。如图 6b 所示，它们在高振荡测试案例中也实现了更高的精度。另一方面，尽管有图 6b 中的结果，但在逼近 (26) 时，[5] 中的 q = 10 q = 10 q=10q=10 规则并没有产生比线性-LG 正交规则和新正交规则更好的结果，这表明在逼近 (25) 时观察到的改进是特定于所考虑的问题或网格的。目前还不清楚为什么一些相同程度的正交规则会产生明显不同的精度水平。研究如何提高正交规则和使用正交规则推导出的算子（如 SBP 导数算子）的精度，可以大幅提高 PDE 数值方法的效率，这也是未来研究的重点。

9. Conclusions 9.结论

The main contribution of this paper is the development of fully-symmetric PI quadrature rules of very high orders. Using a novel approach for generating initial guesses and applying
本文的主要贡献在于开发了高阶全对称 PI 正交规则。采用新颖的方法生成初始猜测并应用
Table 3: Grid convergence study of quadrature approximation errors on tetrahedral meshes.
表 3：四面体网格上正交近似误差的网格收敛研究。
q #elem [6] [5] Line-LG 线路-LG New 新
error 错误 rate 费率 error 错误 rate 费率 error 错误 rate 费率 error 错误 rate 费率
8 6 3 × 6 6 3 × 6 6^(3)xx66^{3} \times 6 5.5338 e − 11 5.5338 e − 11 5.5338 e-115.5338 e-11 7.9018 e − 10 7.9018 e − 10 7.9018 e-107.9018 e-10 1.3511 e − 09 1.3511 e − 09 1.3511 e-091.3511 e-09 1.8845 e − 09 1.8845 e − 09 1.8845 e-091.8845 e-09
7 3 × 6 7 3 × 6 7^(3)xx67^{3} \times 6 1.2141 e − 11 1.2141 e − 11 1.2141 e-111.2141 e-11 9.84 1.5439 e − 10 1.5439 e − 10 1.5439 e-101.5439 e-10 10.59 2.6181 e − 10 2.6181 e − 10 2.6181 e-102.6181 e-10 10.65 3.6643 e − 10 3.6643 e − 10 3.6643 e-103.6643 e-10 10.62
8 3 × 6 8 3 × 6 8^(3)xx68^{3} \times 6 3.2247 e − 12 3.2247 e − 12 3.2247 e-123.2247 e-12 9.93 3.8351 e − 11 3.8351 e − 11 3.8351 e-113.8351 e-11 10.43 6.4692 e − 11 6.4692 e − 11 6.4692 e-116.4692 e-11 10.46 9.0741e-11 10.45
9 3 × 6 9 3 × 6 9^(3)xx69^{3} \times 6 9.9720e-13 9.96 1.1363 e − 11 1.1363 e − 11 1.1363 e-111.1363 e-11 10.33 1.9099e-11 10.36 2.6830 e − 11 2.6830 e − 11 2.6830 e-112.6830 e-11 10.35
9 6 3 × 6 6 3 × 6 6^(3)xx66^{3} \times 6 1.2350 e − 09 1.2350 e − 09 1.2350 e-091.2350 e-09 1.2339 e − 09 1.2339 e − 09 1.2339 e-091.2339 e-09 1.3493 e − 09 1.3493 e − 09 1.3493 e-091.3493 e-09 1.3491 e − 09 1.3491 e − 09 1.3491 e-091.3491 e-09
7 3 × 6 7 3 × 6 7^(3)xx67^{3} \times 6 2.3928 e − 10 2.3928 e − 10 2.3928 e-102.3928 e-10 10.65 2.3914 e − 10 2.3914 e − 10 2.3914 e-102.3914 e-10 10.64 2.6146 e − 10 2.6146 e − 10 2.6146 e-102.6146 e-10 10.64 2.6136 e − 10 2.6136 e − 10 2.6136 e-102.6136 e-10 10.65
8 3 × 6 8 3 × 6 8^(3)xx68^{3} \times 6 5.9118 e − 11 5.9118 e − 11 5.9118 e-115.9118 e-11 10.47 5.9095e-11 10.47 6.4605 e − 11 6.4605 e − 11 6.4605 e-116.4605 e-11 10.47 6.4568 e − 11 6.4568 e − 11 6.4568 e-116.4568 e-11 10.47
9 3 × 6 9 3 × 6 9^(3)xx69^{3} \times 6 1.7452 e − 11 1.7452 e − 11 1.7452 e-111.7452 e-11 10.36 1.7448e - 11 10.36 1.9074 e − 11 1.9074 e − 11 1.9074 e-111.9074 e-11 10.35 1.9060e-11 10.35
10 6 3 × 6 6 3 × 6 6^(3)xx66^{3} \times 6 1.1866 e − 11 1.1866 e − 11 1.1866 e-111.1866 e-11 1.8287 e − 11 1.8287 e − 11 1.8287 e-111.8287 e-11 1.2837 e − 11 1.2837 e − 11 1.2837 e-111.2837 e-11 1.5896 e − 11 1.5896 e − 11 1.5896 e-111.5896 e-11
7 3 × 6 7 3 × 6 7^(3)xx67^{3} \times 6 1.6818 e − 12 1.6818 e − 12 1.6818 e-121.6818 e-12 12.67 2.5810 e − 12 2.5810 e − 12 2.5810 e-122.5810 e-12 12.70 1.8153 e − 12 1.8153 e − 12 1.8153 e-121.8153 e-12 12.69 2.2556 e − 12 2.2556 e − 12 2.2556 e-122.2556 e-12 12.67
8 3 × 6 8 3 × 6 8^(3)xx68^{3} \times 6 3.1724 e − 13 3.1724 e − 13 3.1724 e-133.1724 e-13 12.49 4.8556 e − 13 4.8556 e − 13 4.8556 e-134.8556 e-13 12.51 3.4190 e − 13 3.4190 e − 13 3.4190 e-133.4190 e-13 12.50 4.2581 e − 13 4.2581 e − 13 4.2581 e-134.2581 e-13 12.49
9 3 × 6 9 3 × 6 9^(3)xx69^{3} \times 6 7.3856e - 14 12.37 1.1283 e − 13 1.1283 e − 13 1.1283 e-131.1283 e-13 12.39 7.9528e-14 12.38 9.9177e - 14 12.37
11 6 3 × 6 6 3 × 6 6^(3)xx66^{3} \times 6 1.2702e-12 6.3809 e − 12 6.3809 e − 12 6.3809 e-126.3809 e-12 1.2484 e − 13 1.2484 e − 13 1.2484 e-131.2484 e-13 5.3666e - 12
7 3 × 6 7 3 × 6 7^(3)xx67^{3} \times 6 1.8379e-13 12.54 9.0907e-13 12.64 1.2898 e − 14 1.2898 e − 14 1.2898 e-141.2898 e-14 14.73 7.6052 e − 13 7.6052 e − 13 7.6052 e-137.6052 e-13 12.68
8 3 × 6 8 3 × 6 8^(3)xx68^{3} \times 6 3.5127 e − 14 3.5127 e − 14 3.5127 e-143.5127 e-14 12.39 1.7204 e − 13 1.7204 e − 13 1.7204 e-131.7204 e-13 12.47 1.8509 e − 15 1.8509 e − 15 1.8509 e-151.8509 e-15 14.54 1.4344 e − 13 1.4344 e − 13 1.4344 e-131.4344 e-13 12.49
9 3 × 6 9 3 × 6 9^(3)xx69^{3} \times 6 8.2625e-15 12.29 4.0145 e − 14 4.0145 e − 14 4.0145 e-144.0145 e-14 12.36 3.2092 e − 16 3.2092 e − 16 3.2092 e-163.2092 e-16 14.87 3.3387e - 14 12.38
12 6 3 × 6 6 3 × 6 6^(3)xx66^{3} \times 6 1.6497 e − 14 1.6497 e − 14 1.6497 e-141.6497 e-14 1.0907e-13 1.2484e - 13 1.3055 e − 14 1.3055 e − 14 1.3055 e-141.3055 e-14
7 3 × 6 7 3 × 6 7^(3)xx67^{3} \times 6 1.6983 e − 15 1.6983 e − 15 1.6983 e-151.6983 e-15 14.75 1.1286 e − 14 1.1286 e − 14 1.1286 e-141.1286 e-14 14.72 1.2898 e − 14 1.2898 e − 14 1.2898 e-141.2898 e-14 14.73 1.3540 e − 15 1.3540 e − 15 1.3540 e-151.3540 e-15 14.70
8 3 × 6 8 3 × 6 8^(3)xx68^{3} \times 6 2.4633 e − 16 2.4633 e − 16 2.4633 e-162.4633 e-16 14.46 1.6280 e − 15 1.6280 e − 15 1.6280 e-151.6280 e-15 14.50 1.8509 e − 15 1.8509 e − 15 1.8509 e-151.8509 e-15 14.54 1.9082 e − 16 1.9082 e − 16 1.9082 e-161.9082 e-16 14.67
9 3 × 6 9 3 × 6 9^(3)xx69^{3} \times 6 4.7705 e − 17 4.7705 e − 17 4.7705 e-174.7705 e-17 13.94 3.0184 e − 16 3.0184 e − 16 3.0184 e-163.0184 e-16 14.30 3.2092 e − 16 3.2092 e − 16 3.2092 e-163.2092 e-16 14.88 3.2960 e − 17 3.2960 e − 17 3.2960 e-173.2960 e-17 14.91
q #elem [6] [5] Line-LG New error rate error rate error rate error rate 8 6^(3)xx6 5.5338 e-11 7.9018 e-10 1.3511 e-09 1.8845 e-09 7^(3)xx6 1.2141 e-11 9.84 1.5439 e-10 10.59 2.6181 e-10 10.65 3.6643 e-10 10.62 8^(3)xx6 3.2247 e-12 9.93 3.8351 e-11 10.43 6.4692 e-11 10.46 9.0741e-11 10.45 9^(3)xx6 9.9720e-13 9.96 1.1363 e-11 10.33 1.9099e-11 10.36 2.6830 e-11 10.35 9 6^(3)xx6 1.2350 e-09 1.2339 e-09 1.3493 e-09 1.3491 e-09 7^(3)xx6 2.3928 e-10 10.65 2.3914 e-10 10.64 2.6146 e-10 10.64 2.6136 e-10 10.65 8^(3)xx6 5.9118 e-11 10.47 5.9095e-11 10.47 6.4605 e-11 10.47 6.4568 e-11 10.47 9^(3)xx6 1.7452 e-11 10.36 1.7448e - 11 10.36 1.9074 e-11 10.35 1.9060e-11 10.35 10 6^(3)xx6 1.1866 e-11 1.8287 e-11 1.2837 e-11 1.5896 e-11 7^(3)xx6 1.6818 e-12 12.67 2.5810 e-12 12.70 1.8153 e-12 12.69 2.2556 e-12 12.67 8^(3)xx6 3.1724 e-13 12.49 4.8556 e-13 12.51 3.4190 e-13 12.50 4.2581 e-13 12.49 9^(3)xx6 7.3856e - 14 12.37 1.1283 e-13 12.39 7.9528e-14 12.38 9.9177e - 14 12.37 11 6^(3)xx6 1.2702e-12 6.3809 e-12 1.2484 e-13 5.3666e - 12 7^(3)xx6 1.8379e-13 12.54 9.0907e-13 12.64 1.2898 e-14 14.73 7.6052 e-13 12.68 8^(3)xx6 3.5127 e-14 12.39 1.7204 e-13 12.47 1.8509 e-15 14.54 1.4344 e-13 12.49 9^(3)xx6 8.2625e-15 12.29 4.0145 e-14 12.36 3.2092 e-16 14.87 3.3387e - 14 12.38 12 6^(3)xx6 1.6497 e-14 1.0907e-13 1.2484e - 13 1.3055 e-14 7^(3)xx6 1.6983 e-15 14.75 1.1286 e-14 14.72 1.2898 e-14 14.73 1.3540 e-15 14.70 8^(3)xx6 2.4633 e-16 14.46 1.6280 e-15 14.50 1.8509 e-15 14.54 1.9082 e-16 14.67 9^(3)xx6 4.7705 e-17 13.94 3.0184 e-16 14.30 3.2092 e-16 14.88 3.2960 e-17 14.91| q | #elem | [6] | | [5] | | Line-LG | | New | | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | | | error | rate | error | rate | error | rate | error | rate | | 8 | $6^{3} \times 6$ | $5.5338 e-11$ | | $7.9018 e-10$ | | $1.3511 e-09$ | | $1.8845 e-09$ | | | | $7^{3} \times 6$ | $1.2141 e-11$ | 9.84 | $1.5439 e-10$ | 10.59 | $2.6181 e-10$ | 10.65 | $3.6643 e-10$ | 10.62 | | | $8^{3} \times 6$ | $3.2247 e-12$ | 9.93 | $3.8351 e-11$ | 10.43 | $6.4692 e-11$ | 10.46 | 9.0741e-11 | 10.45 | | | $9^{3} \times 6$ | 9.9720e-13 | 9.96 | $1.1363 e-11$ | 10.33 | 1.9099e-11 | 10.36 | $2.6830 e-11$ | 10.35 | | 9 | $6^{3} \times 6$ | $1.2350 e-09$ | | $1.2339 e-09$ | | $1.3493 e-09$ | | $1.3491 e-09$ | | | | $7^{3} \times 6$ | $2.3928 e-10$ | 10.65 | $2.3914 e-10$ | 10.64 | $2.6146 e-10$ | 10.64 | $2.6136 e-10$ | 10.65 | | | $8^{3} \times 6$ | $5.9118 e-11$ | 10.47 | 5.9095e-11 | 10.47 | $6.4605 e-11$ | 10.47 | $6.4568 e-11$ | 10.47 | | | $9^{3} \times 6$ | $1.7452 e-11$ | 10.36 | 1.7448e - 11 | 10.36 | $1.9074 e-11$ | 10.35 | 1.9060e-11 | 10.35 | | 10 | $6^{3} \times 6$ | $1.1866 e-11$ | | $1.8287 e-11$ | | $1.2837 e-11$ | | $1.5896 e-11$ | | | | $7^{3} \times 6$ | $1.6818 e-12$ | 12.67 | $2.5810 e-12$ | 12.70 | $1.8153 e-12$ | 12.69 | $2.2556 e-12$ | 12.67 | | | $8^{3} \times 6$ | $3.1724 e-13$ | 12.49 | $4.8556 e-13$ | 12.51 | $3.4190 e-13$ | 12.50 | $4.2581 e-13$ | 12.49 | | | $9^{3} \times 6$ | 7.3856e - 14 | 12.37 | $1.1283 e-13$ | 12.39 | 7.9528e-14 | 12.38 | 9.9177e - 14 | 12.37 | | 11 | $6^{3} \times 6$ | 1.2702e-12 | | $6.3809 e-12$ | | $1.2484 e-13$ | | 5.3666e - 12 | | | | $7^{3} \times 6$ | 1.8379e-13 | 12.54 | 9.0907e-13 | 12.64 | $1.2898 e-14$ | 14.73 | $7.6052 e-13$ | 12.68 | | | $8^{3} \times 6$ | $3.5127 e-14$ | 12.39 | $1.7204 e-13$ | 12.47 | $1.8509 e-15$ | 14.54 | $1.4344 e-13$ | 12.49 | | | $9^{3} \times 6$ | 8.2625e-15 | 12.29 | $4.0145 e-14$ | 12.36 | $3.2092 e-16$ | 14.87 | 3.3387e - 14 | 12.38 | | 12 | $6^{3} \times 6$ | $1.6497 e-14$ | | 1.0907e-13 | | 1.2484e - 13 | | $1.3055 e-14$ | | | | $7^{3} \times 6$ | $1.6983 e-15$ | 14.75 | $1.1286 e-14$ | 14.72 | $1.2898 e-14$ | 14.73 | $1.3540 e-15$ | 14.70 | | | $8^{3} \times 6$ | $2.4633 e-16$ | 14.46 | $1.6280 e-15$ | 14.50 | $1.8509 e-15$ | 14.54 | $1.9082 e-16$ | 14.67 | | | $9^{3} \times 6$ | $4.7705 e-17$ | 13.94 | $3.0184 e-16$ | 14.30 | $3.2092 e-16$ | 14.88 | $3.2960 e-17$ | 14.91 |
a node elimination strategy that takes into account the symmetry orbit structures of the optimal quadrature rule estimates, we extend the available set of symmetric PI quadrature rules from degree 50 to 84 on triangles, and from degree 20 to 40 on tetrahedra. Furthermore, the base quadrature rules before node elimination, the line-LG quadrature rules, are efficient in the sense that at very high degrees they require only about 10 % − 20 % 10 % − 20 % 10%-20%10 \%-20 \% and 40 % − 60 % 40 % − 60 % 40%-60%40 \%-60 \% more nodes than estimates of the optimal number of quadrature nodes on triangles and tetrahedra, respectively. Applying node elimination, we derived efficient quadrature rules, most of which have a number of nodes not more than 5 % 5 % 5%5 \% and 25 % 25 % 25%25 \% greater than the lowerbound estimates on triangles and tetrahedra, respectively. Numerical experiments show that the new quadrature rules yield accurate results for highly oscillatory functions and achieve the expected rate of convergence.
考虑到最优正交规则估计的对称轨道结构的节点消除策略，我们将可用的对称 PI 正交规则集从三角形的 50 度扩展到 84 度，从四面体的 20 度扩展到 40 度。此外，节点消除前的基础正交规则（线-LG 正交规则）是高效的，因为在极高的度数下，它们比三角形和正四面体上的最佳正交节点数估计值分别只多出 10 % − 20 % 10 % − 20 % 10%-20%10 \%-20 \% 和 40 % − 60 % 40 % − 60 % 40%-60%40 \%-60 \% 个节点。通过消除节点，我们得出了有效的正交规则，其中大部分规则的节点数分别不超过三角形和正四面体下限估计值的 5 % 5 % 5%5 \% 和 25 % 25 % 25%25 \% 。数值实验表明，新的正交规则对高振荡函数产生了精确的结果，并达到了预期的收敛速度。

Appendix A. Supplementary Materials
附录 A.补充材料

The supplementary material associated with this paper can be found online at https:// github.com/OptimalDesignLab/SummationByParts.jl/tree/master/pi_quadrature_data. It contains both the line-LG quadrature rules and the quadrature rules obtained after application of node elimination.
与本文相关的补充材料可在线查阅：https:// github.com/OptimalDesignLab/SummationByParts.jl/tree/master/pi_quadrature_data。其中包含线性 LG 正交规则和应用节点消除后得到的正交规则。

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URL https://github.com/OptimalDesignLab/SummationByParts.jl
    • Corresponding author: 通讯作者：
    Email addresses: zelalem.worku@mail.utoronto.ca (Zelalem Arega Worku), hickej2@rpi.edu (Jason E. Hicken), dwz@oddjob.utias.utoronto.ca (David W. Zingg)
    电子邮件地址： zelalem.worku@mail.utoronto.ca (Zelalem Arega Worku)、 hickej2@rpi.edu (Jason E. Hicken)、 dwz@oddjob.utias.utoronto.ca (David W. Zingg)