Two-step embedded methods of order s based on s -stage Radau IIA formulas are considered for the variable step-size integration of stiff differential equations. These embedded methods are aimed at local error control and are computed through a linear combination of the internal stages of the underlying method in the last two steps. Particular embedded methods for 2 ≤ s ≤ 7 internal stages with good stability properties and damping for the stiff components are constructed. Furthermore, a new formula for step-size change is proposed, having the advantage that it can be applied to any s -stage Radau IIA method. It is shown through numerical testing on some representative stiff problems that the RADAU5 code by Hairer and Wanner with the new strategy performs as well or even better as with the standard one, which is only feasible for an odd number of stages. Numerical experiments support the efficiency and flexibility of the new step-size change strategy.

中文翻译：

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Radau IIA 方法基于两步的变步长控制

两步嵌入式排序方法s基于s级 Radau IIA 公式被考虑用于刚性微分方程的变步长积分。这些嵌入式方法旨在控制局部误差，并通过最后两个步骤中基础方法的内部阶段的线性组合来计算。2 ≤ 的特殊嵌入方法s构造了≤7个内部级，具有良好的稳定性和刚性组件的阻尼。此外，提出了一种新的步长变化公式，其优点是可以应用于任何s-stage Radau IIA 方法。通过对一些具有代表性的刚性问题的数值测试表明，Hairer 和 Wanner 的 RADAU5 代码采用新策略的性能与标准的一样好，甚至更好，这仅适用于奇数阶段。数值实验支持新的步长变化策略的效率和灵活性。