In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, ch^p+ 1y^(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.

在k步自适应线性多步方法中，系数取决于k -1个最新步长比。以类似的方式，实际和估计的局部误差都将取决于这些步进比。经典误差模型是渐近模型，c h ^{p + 1} y ^{（p + 1）}（t），基于恒定步长分析，其中所有过去的步长同时变为零。这并不反映使用多步方法进行的实际计算，在多步方法中，步长控件会根据先前接受的步长和最近的步长历史记录中的错误信息选择下一步。因此，在可变步长的实现中，即使在渐近状态下，误差模型也必须是动态的并且包括过去的步长比。在本文中，我们得出当地误差及其估计的动态渐进模式，并展示了如何使用动态补偿步长控制器，保持渐近局部误差接近规定的公差TOL。新的错误模型使控制器的使用具有更高的稳定性，从而产生了更规则的步长序列。数值示例说明了动态补偿控制的影响，并且误差估计器的正确选择会影响效率。