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：摘要：我们提出高效算法，用于对解可表示为三角矩阵函数的大规模振荡型二阶常微分方程组进行时间积分。这些算法基于二阶常微分方程的残差概念，从而将“残差-时间重启”Krylov子空间框架——该框架近期被引入用于一阶常微分方程时间积分中出现的指数函数和φ函数——扩展至我们的场景。进而证明，在Gautschi余弦格式中采用我们的重启方法，可在许多情况下进一步降低计算成本。我们利用Faber级数与Chebyshev级数分析残差收敛性，并通过数值实验补充理论结果，展示所提方法的效率。