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）是一类用于求解常微分方程数值解的迭代方法。SDC可被解释为求解全隐式配置问题的Picard迭代，并通过低阶方法进行预处理。该方法已在一阶问题中得到广泛研究，常使用显式、隐式或隐式-显式欧拉方法及其他低阶方法作为预处理器。对于一阶问题，SDC能达到任意阶精度且具有良好的稳定性。尽管已有将SDC应用于二阶洛伦兹方程的数值结果，但尚无关于SDC用于二阶问题的理论分析。本文以速度-Verlet方法为基础方法，针对一般二阶初值问题，分析了SDC的收敛性和稳定性。分析表明，收敛阶次取决于系统中力是否依赖速度。我们还证明了在一定条件下SDC迭代的稳定性。最后，数值结果表明，SDC在计算效率上优于简单Picard迭代或四阶Runge-Kutta-Nyström方法。