一种基于加权残值法的高阶辛算法

A High Order Symplectic Algorithm Based on Weighted Residual Method

提出了利用加权残值法构造高阶辛算法的一种新途径。首先根据加权残值法的思想,在时间子域内给出了哈密顿正则方程伽辽金法所对应的积分方程,然后在该时间子域内采用相同的拉氏插值作为位移和动量的试函数,并将这些试函数代入到积分方程中,通过数值积分,将原动力学初值问题转为以插值点位移和动量为未知量的代数方程组。对于非线性问题,给出了一种能显著提高牛顿迭代法计算效率的初值选取方案。最后,对算法的保辛性和性能进行了详细的讨论。通过与同阶辛RK法相比较,两种方法精度几乎完全相同,但文中方法更简便,计算量更小。数值算例结果表明该法在计算精度和效率上均具有良好的性能。

A new way to construct high order symplectic algorithms is proposed based on weighted residual method. Firstly,in the time subdomain,the corresponding integral equation of Galerkin method for Hamilton dual equation based on the idea of weighted residual method is proposed,then the generalized displacement and momentum are approximated by the same Lagrange interpolation within the time subdomain,which are substituted into the corresponding integral equation. By numerical integration,the original initial value problem of dynamics is expressed as algebraic equations with displacement and momentum at the interpolation points as unknown variables. For nonlinear dynamic systems,a simple scheme of choosing initial values,which can significantly improve the computational efficiency for Newton-Raphson method,is presented. Finally,the symplecticity and performance of the proposed algorithms are discussed in detail. Compared with the same order symplectic Runge-Kutta methods,the accuracy of the two methods are almost the same,but the proposed algorithms are much simpler and less computational expense. The numerical results illustrate that the proposed algorithms show good performance in accuracy and efficiency.