Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order p and stage order $q=p$, some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.

中文翻译：

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常微分方程的可变步长 SDIMSIM

二阶导数一般线性方法 (SGLM) 已经使用 Nordsieck 技术在可变步长环境中实现。在本文中，我们直接在非均匀网格上引入了可变步长 SGLM。通过推导所提出的阶p和阶段阶方法的阶条件$q=磷$，给出了这些方法的一些明确的例子，直到四阶。通过一些数值实验，我们展示了所提出的方法解决非刚性问题的效率，并确认了理论收敛顺序。