Analysis for parareal algorithms applied to Hamiltonian differential equations
Analysis for parareal algorithms applied to Hamiltonian differential equations
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DOI：
10.1016/j.cam.2013.01.011
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发表时间：
2014-03
期刊：
影响因子：
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通讯作者：
M. Gander;E. Hairer
中科院分区：
文献类型：
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作者：
M. Gander;E. Hairer
长时间积分是哈密顿系统数值求解中的一个重要问题。它们是耗时的，并且出于效率的原因考虑使用并行架构是很自然的。在这种背景下，Parareal算法已被一些作者提出。本工作是理论研究的Parareal算法时，它应用于Hamilton微分方程。向后误差分析的思想被用来深入了解数值近似的长时间行为。其中一个主要结果是，收敛的parareal迭代限制的时间窗口的长度。对于近似可积系统，它的长度由粗积分器的精度的倒数的平方根所限定。数值实验证实了理论界。
Long-time integrations are an important issue in the numerical solution of Hamiltonian systems. They are time consuming and it is natural to consider the use of parallel architectures for reasons of efficiency. In this context the parareal algorithm has been proposed by several authors.The present work is a theoretical study of the parareal algorithm when it is applied to Hamiltonian differential equations. The idea of backward error analysis is employed to get insight into the long-time behavior of numerical approximations. One of the main results is that convergence of the parareal iterations restricts the length of the time window. For nearly integrable systems its length is bounded by the square root of the inverse of the accuracy of the coarse integrator. The theoretical bounds are confirmed by numerical experiments.