We propose algorithms for efficient time integration of large systems of oscillatory second order ordinary differential equations (ODEs) whose solution can be expressed in terms of trigonometric matrix functions. Our algorithms are based on a residual notion for second order ODEs, which allows to extend the ``residual-time restarting'' Krylov subspace framework -- which was recently introduced for exponential and $\varphi$-functions occurring in time integration of first order ODEs -- to our setting. We then show that the computational cost can be further reduced in many cases by using our restarting in the Gautschi cosine scheme. We analyze residual convergence in terms of Faber and Chebyshev series and supplement these theoretical results by numerical experiments illustrating the efficiency of the proposed methods.
翻译:我们建议采用算法,将大型的随机第二顺序普通差分方程式系统(ODEs)高效的时间整合到我们的设置中,这些系统的解决办法可以表现为三角矩阵功能。我们的算法基于第二顺序的剩余概念,它允许扩展“剩余时间重新启动”Krylov子空间框架,该框架是最近为指数值和在时间整合第一顺序的ODEs中出现的美元功能而引入的。然后我们表明,在许多情况下,计算成本可以通过在Gautschi comsine计划中重新启用来进一步降低。我们分析了Faber和Chebyshev系列的剩余趋同,并通过数字实验来补充这些理论结果,说明拟议方法的效率。