In this paper, the implementation of second derivative general linear methods (SGLMs) in a variable stepsize environment using Nordsieck technique is discussed and various implementation issues are investigated. All coefficients of a method of order four together with its error estimate are obtained. The method is derived with the aim of good zero-stability properties for a large range of ratios of sequential stepsizes to implement in a variable stepsize mode. The numerical experiments indicate that the constructed error estimate is very reliable in a variable stepsize environment and beautifully confirm the efficiency and robustness of the proposed scheme based on SGLMs. Moreover, the results verify that the proposed scheme outperforms the code ode15s from Matlab ODE suite on systems whose Jacobian has eigenvalues which are close to the imaginary axis.

中文翻译：

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二阶导数一般线性方法的实现

本文讨论了使用Nordsieck技术在变步长大小环境中实现二阶导数一般线性方法（SGLM）的方法，并研究了各种实现问题。获得了四阶方法的所有系数及其误差估计。该方法的目的是为了在可变步长模式下实现大范围的连续步长比比率时具有良好的零稳定性。数值实验表明，所构造的误差估计在可变步长的环境中是非常可靠的，并且很好地证实了基于SGLM的方案的有效性和鲁棒性。此外，结果验证了该方案优于Matlab的代码ode15s。 ODE套件，用于其Jacobian特征值接近虚轴的系统。