A Class of Multirate Infinitesimal GARK Methods
A Class of Multirate Infinitesimal GARK Methods
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一类多速率无穷小GARK方法
DOI：
10.1137/18m1205492
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Adrian Sandu
中科院分区：
文献类型：
--
作者：
Adrian Sandu
在许多实际应用中出现的微分方程具有多时间尺度的特点。多速率时间积分试图通过用不同的、适当的时间步长离散每个尺度来有效地解决这些问题，同时确保数值解的整体精度和稳定性。Knoth和Wolke(APNUM，1998)在一篇开创性的论文中提出了一种混合求解方法：用显式Runge-Kutta方法离散慢分量，通过修正的快速微分方程推进快分量。这一思想导致了Wensch等人开发多速率无穷小步长(MIS)方法。(BIT，2009)Gunther和Sandu(Bit，2016)将管理信息系统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分量的数值解过程方面提供了极大的灵活性。
本文构造了一族多速率无穷小Gark格式(MRI-GARK)，它以多种方式扩展了混合动力学方法。发展了阶数条件理论和稳定性分析，构造了四阶以下的实用显式和隐式方法。数值结果证实了理论结果。我们期望新的MRI-GARK族最适用于时间尺度完全不同的方程组，其中快过程是弥散的，而快分量对慢动力学的影响很弱。
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.)Gunther 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.
电路与分布式器件耦合问题的动态迭代
DOI：
10.1137/120867111
发表时间：
2013
期刊：
SIAM J. Sci. Comput.
影响因子：
--
作者：
Bartel;Günther;Schöps S.
通讯作者：
Schöps S.