一类多速率无穷小GARK方法
A Class of Multirate Infinitesimal GARK Methods
AI总结：
本文提出了一类多速率无穷小GARK方法，扩展了混合动力学方法，通过发展阶条件理论和稳定性分析，构造了最高四阶的显式和隐式方法，并通过数值结果验证了理论发现，该方法适用于具有广泛不同时刻尺度的方程组，特别是当快速过程是分散的且对慢动态影响较小时。
摘要：
在许多实际应用中出现的微分方程具有多个时间尺度。多速率时间积分通过用不同的适当时间步长对每个尺度进行离散化，以高效地解决这些问题，同时确保数值解的整体精度和稳定性。在一篇开创性的论文中，Knoth和Wolke（APNUM，1998）提出了一种混合解决方案方法：用显式龙格-库塔方法离散化慢成分，并通过修改后的快速微分方程推进快速成分。这一想法促成了Wensch等人（BIT，2009）提出的多速率无穷小步（MIS）方法。Günther和Sandu（BIT，2016）将MIS方案解释为多速率通用结构加法龙格-库塔（MR-GARK）方法的特例。混合方法在选择快速成分的数值求解过程时提供了极高的灵活性。本文构建了一类多速率无穷小GARK方案（MRI-GARK），以多种方式扩展了混合动力学方法。发展了阶条件理论和稳定性分析，并构造了最高四阶的显式和隐式方法。数值结果验证了理论发现。我们预计新的MRI-GARK家族将最适用于具有广泛不同时刻尺度的方程组，其中快速过程是分散的，且快速成分对慢动态的影响较弱。
英文摘要：
Differential equations arising in many practical applications are characterized by multiple time scales. Multirate time integration seeks to solve them efficiently by discretizing each scale with a different, appropriate time step, while ensuring the overall accuracy and stability of the numerical solution. In a seminal paper Knoth and Wolke (APNUM, 1998) proposed a hybrid solution approach: discretize the slow component with an explicit Runge-Kutta method, and advance the fast component via a modified fast differential equation. The idea led to the development of multirate infinitesimal step (MIS) methods by Wensch et al. (BIT, 2009.)Günther and Sandu (BIT, 2016) explained MIS schemes as a particular case of multirate General-structure Additive Runge-Kutta (MR-GARK) methods. The hybrid approach offers extreme flexibility in the choice of the numerical solution process for the fast component. This work constructs a family of multirate infinitesimal GARK schemes (MRI-GARK) that extends the hybrid dynamics approachin multiple ways. Order conditions theory and stability analyses are developed, and practical explicit and implicit methods of up to order four are constructed. Numerical results confirm the theoretical findings. We expect the new MRI-GARK family to be most useful for systems of equations with widely disparate time scales, where the fast process is dispersive, and where the influence of the fast component on the slow dynamics is weak.
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