SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3082-A3113, January 2021.
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. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called infinitesimal formulation for the fast time scale through definition of a sequence of modified “fast" initial-value problems that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.

中文翻译：

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隐-显多速率无穷小 GARK 方法

SIAM 科学计算杂志，第 43 卷，第 5 期，第 A3082-A3113 页，2021 年 1 月。
这项工作的重点是开发一类新的高阶精确方法，用于常微分方程系统的多速率时间积分。与该领域的其他近期工作不同，所提出的方法支持对慢时间尺度的混合隐式-显式 (IMEX) 处理。除了允许这种缓慢的时间尺度灵活性外，所提出的方法通过定义一系列修改过的“快速”初始值问题，可以使用任何可行的算法来解决快速时间尺度的所谓无穷小公式。我们命名建议的类作为隐式-显式多速率无穷小广义结构加性 Runge--Kutta (IMEX-MRI-GARK) 方法。除了定义这些方法，我们证明它们可以被视为 GARK 方法的特定实例，并在 IMEX-MRI-GARK 系数上推导出一组阶条件，以保证整体多速率方法的三阶和四阶精度。此外，我们提供了三种特定的 IMEX-MRI-GARK 方法，两种三阶方法和一种四阶方法。我们最后对两个多速率测试问题进行了数值模拟，展示了这些方法的预测收敛率，并将它们的效率与传统的 IMEX 多速率方案和最近的三阶和四阶隐式 MRI-GARK 方法进行了比较。