任意阶帕德封闭锚定二阶导数时间离散化：\(s\) 个活跃阶段，\(2s\) 阶，以及 \(L\) 稳定性
Arbitrary-Order Padé-Closed Anchored Two-Derivative Time Discretizations: $s$ Active Stages, Order $2s$, and $L$-Stability
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中文总结 AI 辅助
研究构建任意阶隐式二阶导数单步方法，利用向量场及其导数，通过埃尔米特矩条件和帕德逼近确定系数，具有 \(L\) 稳定性，经符号验证和计算确认高阶精度，与其他方法比较显示在宽步长范围有优势。
中文摘要 AI 辅助
本文在锚定活跃阶段公式中构建了一族任意阶的隐式二阶导数单步方法。该方法在每个信息节点使用向量场及其一阶全时间导数，在不增加未知阶段状态数量的情况下丰富局部埃尔米特数据。以已知初始值为锚点，\(s\) 个未知活跃阶段，通过 \(2s\) 个埃尔米特矩条件得到全局 \(2s\) 阶。每个阶段行的其余两个系数由次对角线二阶帕德逼近 \([s - 1/s + 1]_{e^z}\) 确定。帕德 - 埃尔米特基定理证明了对于每个有序实节点集，封闭是唯一的，保留所有矩条件，并给出 \(\det(I - zA - z^2\widehat{A}) = Q_s(z)\) 和 \(R_s(z) = P_s(z)/Q_s(z)\)。因此耦合阶段系统没有隐藏极点，且对于每个正整数 \(s\)，接受的单步映射是 \(L\) 稳定（因此是 \(A\) 稳定）的。通过 \(s = 6\) 进行了精确符号验证，高精度计算确认了前四个成员的 \(2\)、\(4\)、\(6\) 和 \(8\) 阶。在活跃阶段数量相等时，与高斯 - 勒让德和拉道 IIA 方法的比较表明，该构造在宽步长范围内具有高阶精度和强刚性阻尼的组合。
英文摘要
An arbitrary-order family of implicit two-derivative one-step methods is constructed in an anchored active-stage formulation. At each information node the method uses both the vector field and its first total time derivative, enriching the local Hermite data without increasing the number of unknown stage states. With the known initial value retained as an anchor and $s$ unknown active stages, $2s$ Hermite moment conditions yield global order $2s$. The two remaining coefficients in each stage row are fixed by the second-subdiagonal Padé approximant $[s-1/s+1]_{e^z}$. For every ordered real node set, a Padé--Hermite basis theorem proves that the closure is unique, preserves all moment conditions, and gives $\det(I-zA-z^2\widehat A)=Q_s(z)$ and $R_s(z)=P_s(z)/Q_s(z)$. Hence the coupled stage system has no hidden poles and the accepted one-step map is $L$-stable (and therefore $A$-stable) for every positive integer $s$. Exact symbolic verification is reported through $s=6$, and high-precision computations confirm orders $2,4,6,$ and $8$ for the first four members. At equal active-stage count, comparisons with Gauss--Legendre and Radau IIA methods demonstrate the combined high-order accuracy and strong stiff damping of the construction over broad step-size ranges.
