Euler—Lagrange方程的高精度计算方法

Euler—Lagrange方程是多体系统动力学的基本方程之一，是高指标的强非线性微分代数方程组。利用零空间方法对Euler—Lagrange方程作简化处理，然后利用高精度谱积分对得到的微分代数方程组作数值离散，形成配置离散格式。针对高阶微分代数方程的离散方程组的病态问题，采用预条件技术改善了方程组的求解条件，然后利用Newton—Krylov方法迭代求解。这种求解技术可以得到任意阶精度且A-稳定算法，并且采用预条件技巧极大的降低了计算的复杂性。

A Algorithm of High Precision for Euler-Lagrange Equations

Euler-Lagrange equation, a fundamental equation widely used in scientific study and technical problems, is a large nonlinear high index DAE. First, the Euler-Lagrange equation is simplified by null space technique. Second, by using a high precise spectral integral and a numerical method of high precision, the Euler-I.agrange equation is discretized into a collocation formula. Since the discretized equation systems of high index DAE is an ill-conditioned problem, preconditioning techniques are adopted to improve the condition of the systems. Newton-Krylov schemes is applied to solve the systems. The new algorithm is A-stable, can reach machine accuracy, and requires modest computational expenses due to our preconditioned techniques.