SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1609-A1638, January 2020.
Traditional time discretization methods use a single timestep for the entire system of interest and can perform poorly when the dynamics of the system exhibits a wide range of time scales. Multirate infinitesimal step (MIS) methods [O. Knoth and R. Wolke, Appl. Numer. Math., 28 (1998), pp. 327--341] offer an elegant and flexible approach to efficiently integrate such systems. The slow components are discretized by a Runge--Kutta method, and the fast components are resolved by solving modified fast differential equations. Sandu [SIAM J. Numer. Anal., 57 (2019), pp. 2300--2327] developed the multirate infinitesimal general-structure additive Runge--Kutta (MRI-GARK) family of methods that includes traditional MIS schemes as a subset. The MRI-GARK framework allowed the construction of the first fourth order MIS schemes. This framework also enabled the introduction of implicit methods, which are decoupled in the sense that any implicitness lies entirely within the fast or slow integrations. It was shown by Sandu that the stability of decoupled implicit MRI-GARK methods has limitations when both the fast and slow components are stiff and interact strongly. This work extends the MRI-GARK framework by introducing coupled implicit methods to solve stiff multiscale systems. The coupled approach has the potential to considerably improve the overall stability of the scheme, at the price of requiring implicit stage calculations over the entire system. Two coupling strategies are considered. The first computes coupled Runge--Kutta stages before solving a single differential equation to refine the fast solution. The second alternates between computing coupled Runge--Kutta stages and solving fast differential equations. We derive order conditions and perform the stability analysis for both strategies. The new coupled methods offer improved stability compared to the decoupled MRI-GARK schemes. The theoretical properties of the new methods are validated with numerical experiments.

中文翻译：

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具有多个时间尺度的刚性系统的耦合多速率无穷GARK方案

SIAM科学计算杂志，第42卷，第3期，第A1609-A1638页，2020年1月。
传统的时间离散化方法对整个目标系统使用单个时间步长，并且当系统的动力学表现出较大的时间范围时，其性能可能会很差。多速率无穷小步长（MIS）方法[O. Knoth和R.Wolke，应用 Numer。Math。，28（1998），pp。327--341]提供了一种优雅而灵活的方法来有效地集成此类系统。慢速分量通过Runge-Kutta方法离散化，快速分量通过求解修正的快速微分方程来求解。Sandu [SIAM J. Numer。Anal。，57（2019），pp。2300--2327]开发了多速率无穷小通用结构加法Runge-Kutta（MRI-GARK）系列方法，其中包括传统的MIS方案作为子集。MRI-GARK框架允许构建第一个四阶MIS方案。该框架还允许引入隐式方法，这些隐式方法在任何隐含性完全位于快速或慢速集成范围内的意义上是分离的。Sandu证明，当快速分量和慢速分量都刚性且相互作用强烈时，解耦隐式MRI-GARK方法的稳定性受到限制。这项工作通过引入耦合隐式方法来解决刚性多尺度系统，扩展了MRI-GARK框架。耦合方法有可能显着提高该方案的整体稳定性，但代价是需要对整个系统进行隐式阶段计算。考虑了两种耦合策略。在求解单个微分方程以完善快速求解之前，第一个计算耦合的Runge-Kutta级。第二步在计算耦合的Runge-Kutta级和求解快速微分方程之间交替进行。我们导出订单条件并执行两种策略的稳定性分析。与解耦的MRI-GARK方案相比，新的耦合方法提供了更高的稳定性。数值实验验证了新方法的理论特性。