发表机构
Michigan State University（密歇根州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一类标量常微分方程的单步隐式方法，利用缩放角均值更新，通过选择缩放因子可实现四阶精度，数值实验显示其同步长下误差小于经典四阶Runge--Kutta方法。
AI 中文摘要
本文提出了一类用于求解标量常微分方程的单步隐式方法。每一步的更新采用两次向量场评估的缩放角均值，且正缩放因子可随步变化。局部截断误差的主导项可表示为缩放解图的带符号曲率导数。我们证明，当解图在固定正缩放下具有恒定带符号曲率且每次隐式更新唯一时，该方法在网格点处可重现精确解。当不满足该特殊几何条件时，我们建立了正缩放因子序列满足适当一致条件下的二阶相容性与收敛性。此外，通过选择缩放因子抵消局部截断误差中相关曲率项，可实现三阶和四阶相容性与收敛性。仅利用给定问题数据，我们分类了这些改进的可能性并确定了对应缩放因子的选择。一个示例表明，在相同步长下，所提出的四阶方法比经典四阶Runge--Kutta方法产生的误差更小。
英文摘要
This paper introduces a family of one-step implicit methods for solving scalar ordinary differential equations. At each step, the update uses a scaled angular mean of two vector-field evaluations, and the positive scale may vary from step to step. The leading terms of the local truncation error can be expressed in terms of the derivative of the signed curvature of the scaled solution graph. We prove that the proposed method reproduces the exact solution at the mesh points when the solution graph has constant signed curvature under a fixed positive scaling and each implicit update is unique. When this special geometric condition is not satisfied, we establish second-order consistency and convergence for positive scale sequences satisfying suitable uniform conditions. Furthermore, third- and fourth-order consistency and convergence can be achieved by choosing the scale to cancel the relevant curvature terms in the local truncation error. Using only the given problem data, we classify when these improvements are possible and determine the corresponding scale choices. An example shows that the proposed fourth-order method can yield smaller errors than the classical fourth-order Runge--Kutta method at the same step size.
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