This paper aims to study a new reproducing kernel (RK) function-based collocation method for nonlinear Hamiltonian systems. By applying the associated functions of RK spaces $W_{2,0}^2$, $W_{2,0}^3$ and $W_{2,0}^4$, we propose the second, third and fourth-order schemes, respectively. Since the coefficient matrix of the linear system obtained by our scheme is symmetric and positive definite, our approach is uniquely solvable.The numerical experiments verify that our algorithms are efficient and can simulate the long time behavior including energy conservation and symplectic structure properties.

中文翻译：

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求解非线性哈密顿系统的一种新的分段再现核函数算法

本文旨在研究一种新的基于再生核（RK）函数的非线性哈密顿系统搭配方法。通过应用 RK 空间的相关函数$W_{2,0}^2$,$W_{2,0}^3$和$W_{2,0}^4$，我们分别提出二阶、三阶和四阶方案。由于我们的方案得到的线性系统的系数矩阵是对称的并且是正定的，所以我们的方法是唯一可解的。数值实验验证了我们的算法是有效的，并且可以模拟包括能量守恒和辛结构性质在内的长时间行为。