Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta-Nystr\"om method.

翻译：谱延迟校正（SDC）是一类用于常微分方程数值求解的迭代方法。SDC可理解为求解全隐式配置问题的皮卡迭代，并以低阶方法作为预处理器。该方法已在利用显式、隐式或隐式-显式欧拉法及其他低阶方法作为预处理器的前阶问题中得到广泛研究。针对前阶问题，SDC可实现任意阶精度并具有良好的稳定性特征。尽管已有将SDC应用于二阶洛伦兹方程的数值结果，但目前尚无关于SDC应用于二阶问题的理论分析。本文以速度Verlet算法作为一般二阶初值问题的基方法，对SDC的收敛性和稳定性进行了分析。我们的分析证明了收敛阶取决于系统中力是否依赖于速度，同时表明SDC迭代在特定条件下具有稳定性。最后，我们证明了SDC在计算效率上可优于简单皮卡迭代或四阶龙格-库塔-尼斯特伦方法。