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 (Knoth and Wolke, 1998) 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 (2018) 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.
翻译:传统时间离散方法对整个利益体系使用单一时间步,当系统动态显示具有广泛时间尺度时,效果可能很差。多端无限级步骤(MIS)方法(Knoth和Wolke,1998年)为高效整合这些体系提供了优雅和灵活的方法。慢度组件通过龙格-库塔方法分解,快速组件通过解决经修改的快速差异方程式来解决。桑杜(2018年)开发了多端通用结构(多端结构)通缩-龙格-库特塔(MRI-GARK)系列方法,其中包括传统的MIS计划作为子阶段。MRI-GARK框架允许构建第四种阶梯级的MIS(MIS)方法。这个框架还使得引入了隐性方法,因为任何隐性部分都完全存在于快速或缓慢的整合中。桑杜证明,当快速和缓慢的MRI-GARK(M-GARK)战略的精细度分解和互动性能强时,这种方法的稳定性会限制。这项工作将整个MARK框架扩展为双端级的运行阶段,同时引入了双级的计算方法,在快速级计算方法中,在快速级平流法中,在快速系统上将稳定度分析中将快速级规则化方法进行。