论文标题
Krylov子空间残留和重新启动某些二阶微分方程
Krylov subspace residual and restarting for certain second order differential equations
论文作者
论文摘要
我们提出了算法，以有效地整合大型振荡二阶的普通微分方程（ODE）的算法，其解决方案可以用三角矩阵函数表示。我们的算法基于二阶频率的残留概念，该概念允许将``剩余时间重新启动'''Krylov子空间框架（最近引入了指数和$φ$函数在一阶ODE的时代集成中发生的$φ$函数）。然后，我们证明，通过在Gautschi余弦方案中使用我们的重新启动，可以在许多情况下进一步降低计算成本。我们根据Faber和Chebyshev系列分析了剩余收敛性，并通过说明所提出方法的效率来补充这些理论结果。
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 $φ$-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.
