基于截断梯形法则的卷积求积
Convolution quadrature based on a truncated trapezoidal rule
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中文总结 AI 辅助
研究截断梯形法则这一A稳定二阶多步法，得到使主误差常数最小化的系数闭式及误差常数公式，将其用于卷积求积框架，相比梯形法则有更温和要求，数值实验显示其在符号扰动下稳定且误差常数比BDF2小。
中文摘要 AI 辅助
我们研究截断梯形法则，它是一族由整数\(J \geq 2\)参数化的A稳定二阶多步法，是BDF2和梯形法则之间的一种折衷。我们得到了在A稳定性约束下使主误差常数最小化的系数的闭式表达式，并推导出了相应主误差常数的显式公式。随着\(J\)增加，后者减小到最优的达赫奎斯特值\(1/12\)，并且对于每个\(J \geq 2\)都严格小于BDF2常数。我们在卷积求积框架内应用截断梯形法则。它在闭单位圆盘邻域内的解析性产生了比梯形法则更温和的正则性和扰动要求，同时通过增加\(J\)误差常数可以任意接近最优值。数值实验表明，基于截断梯形法则的卷积求积在符号扰动下保持稳定（梯形法则在此处失效），同时实现比BDF2更小的误差常数。
英文摘要
We study the truncated trapezoidal rule, a family of A-stable second order multistep methods parametrized by an integer $J \ge 2$, a compromise between BDF2 and the trapezoidal rule. We obtain a closed-form expression for the coefficients that minimize the principal error constant under the A-stability constraint and we derive an explicit formula for the corresponding principal error constant. The latter decreases to the optimal Dahlquist value $1/12$ as $J$ increases, and is strictly smaller than the BDF2 constant for every $J \ge 2$. We apply the truncated trapezoidal rule within the convolution quadrature framework. Its analyticity in a neighbourhood of the closed unit disk yields milder regularity and perturbation requirements than those of the trapezoidal rule, while the error constant can be made arbitrarily close to the optimal one by increasing $J$. Numerical experiments show that convolution quadrature based on the truncated trapezoidal rule remains stable under symbol perturbations, where the trapezoidal rule fails, while achieving a smaller error constant than BDF2.
