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（Spectral Deferred Corrections, SDC）是一类用于求解常微分方程的迭代数值方法。SDC可解释为对完全隐式配置问题的Picard迭代，并采用低阶方法作为预处理器。该方法已在显式/隐式/隐式-显式欧拉等低阶格式作为预处理器的情形下，针对一阶问题得到广泛研究。对于一阶问题，SDC具有任意阶精度和良好的稳定性。尽管已有将SDC应用于二阶洛伦兹方程的数值算例，但尚未有关于SDC应用于二阶问题的理论成果。本文以速度-Verlet法为基础方法，对一般二阶初值问题的SDC收敛性与稳定性开展分析。分析表明：收敛阶数取决于系统中力是否依赖于速度；同时证明SDC迭代在特定条件下具有稳定性。最后，我们证明SDC在计算效率上优于简单Picard迭代或四阶Runge-Kutta-Nyström方法。