利用分段算法微分整合Lipschitz动力系统
Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation
AI总结：
本文提出一种改进的梯形法则，用于求解具有分段光滑右端函数的初值问题，解决了传统方法在解轨迹穿过非光滑点时精度下降的问题，并实现了更高的收敛阶、能量守恒性和插值多项式。
摘要：
本文分析了一种广义梯形法则，用于求解具有分段光滑右端函数$F:\R^n\to\R^n$的初值问题。当应用于此类问题时，传统梯形法则在解轨迹穿过$F$的非光滑点时会损失精度。所提出的广义梯形法则有三方面优势：首先，可以实现比传统方法更高的收敛阶；其次，该方法对分段线性哈密顿系统具有能量守恒性；最后，类似于传统情况，我们推导了数值轨迹的三次插值多项式。在光滑情况下，广义规则退化为传统规则。因此，它是对传统理论的适当扩展。给出了误差估计器并展示了数值结果。
英文摘要：
In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side \(F:\R^n\to\R^n\). When applied to such a problem the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of \(F\). The advantage of the proposed generalized trapezoidal rule is threefold: Firstly we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third order interpolation polynomial for the numerical trajectory. In the smooth case the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
