This work considers multirate generalized-structure additively partitioned Runge–Kutta methods for solving stiff systems of ordinary differential equations with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

这项工作考虑了多速率广义结构可加分配的Runge-Kutta方法，用于求解具有多个时间尺度的常微分方程的刚性系统。与整体式单速率方法相比，这些方法以不同的时间步长处理系统的不同分区，从而获得更具针对性和效率的解决方案。通过在所有分区上使用隐式方法，方法必须在稳定性和求解阶段非线性方程式的成本之间找到平衡。为了表征这一重要的折衷，我们探索了多速率耦合策略，评估线性稳定性的问题以及有效地实现阶段方程的牛顿迭代的技术。与许多现有的多速率稳定性分析（仅限于特定方法）不同，我们提供有关稳定性的一般说明，并描述某些类型的多速率方案的基本局限性。推导了新的隐式多速率方法直至四阶，并通过数值测试验证了它们的准确性和效率性质。