SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1245-A1268, January 2020.
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriate form, we derive a multirate algorithm to approximate the action of the matrix exponential through the definition of modified “fast” initial-value problems. These fast problems may be solved using any viable solver, enabling multirate simulations through use of a subcycled method. Due to this structure, we name these multirate exponential Runge--Kutta (MERK) methods. In addition to showing how MERK methods may be derived, we provide rigorous convergence analysis, showing that for an overall method of order $p$, the fast problems corresponding to internal stages may be solved using a method of order $p-1$, while the final fast problem corresponding to the time-evolved solution must use a method of order $p$. Numerical simulations are then provided to demonstrate the convergence and efficiency of MERK methods with orders three through five on a series of multirate test problems.

中文翻译：

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一类新的多速率微分方程高阶方法

SIAM科学计算杂志，第42卷，第2期，第A1245-A1268页，2020年1月。
这项工作的重点是为一类常微分方程系统的多速率时间积分开发一类新的高阶精确方法。所提出的方法基于显式单步指数积分器的特定子集。更准确地说，从适当形式的显式指数Runge-Kutta方法开始，我们通过定义修改后的“快速”初值问题，推导了一种多速率算法来近似矩阵指数的作用。可以使用任何可行的求解器解决这些快速问题，从而通过使用子循环方法实现多速率仿真。由于这种结构，我们将这些多速率指数Runge-Kutta（MERK）方法命名。除了说明如何推导MERK方法外，我们还提供严格的收敛分析，表明对于阶次为$ p $的整体方法，可以使用阶次为$ p-1 $的方法来解决与内部阶段相对应的快速问题，而与时间演化解相对应的最终快速问题必须采用以下方法：订购$ p $。然后提供了数值模拟，以证明在一系列多速率测试问题上，MERK方法的收敛性和效率为三到五阶。