基于非传统哈密顿变分原理的高阶辛算法

A high order symplectic algorithm based on unconventional Hamilton variational principle

给出了非传统哈密顿变分原理的一种简化形式,并在此基础上利用拉格朗日多项式近似位移和动量,采用高斯积分法对时间积分,建立了针对动力学初值问题的一类高阶辛算法。在建立高阶辛算法的过程中,本文方法与基于传统哈密顿变分原理的辛算法不同,无需由端值问题向初值问题转换,因此更加简捷有效。此外,给出了线性动力问题中本文算法保辛性的证明。当位移、动量的插值次数和高斯积分点个数均为m时,本文算法是具有2m阶精度的辛算法,且是线性无条件稳定的。通过数值算例结果表明,本文算法与辛算法性质吻合,并且计算效率比同阶辛龙格库塔法提高了约50%。

A simplified form of unconventional Hamilton variational principle is presented. Based on this variational principle, the generalized displacements and momenta are approximated by the Lagrange interpolation within the time step. After numerical integration and variational operation, a high order symplectic algorithm corresponding to initial value problems of dynamics is constructed. Different from the algorithm based on Hamilton variational principle, the proposed algorithm needs not to convert the boundary value problems into initial value problems, which is simpler and convenient. Furthermore, the proof for symplecticity of the proposed algorithm in linear dynamic system is given. And when the orders of the polynomials for displacements and momenta and numbers of Gauss integration point are all m, the proposed method is a symplectic algorithm with 2m order accuracy and linear unconditionally stable. Finally, numerical examples indicate that the proposed algorithm has the same properties as the symplectic algorithms, and improves the computational efficiency about by 50% compared with the same order symplectic Runge-Kutta methods.