给出一种求解在长波近似条件下水波问题所对应的哈密顿系统的辛几何算法. 首先将生成函数法推广至无穷维哈密顿系统; 然后,基于无穷维系统自身的哈密顿函数,而不是其有限维近似系统的哈密顿函数,构造辛差分格式;最后,用空间离散的辛格式实现仿真计算. 与非辛算法相比,该辛算法在长时间仿真中能给出稳定的数值结果. 与传统的求解无穷维哈密顿系统的辛几何算法相比,该算法计算效率更高,其仿真结果更准确.
An symplectic algorithm is presented for solving the Hamiltonian systems of the water-wave problem under long-wave approximation. Firstly, the generating function method is generalized to infinite-dimensional Hamiltonian systems, and then symplectic schemes are deduced directly from the Hamiltonian function of infinite-dimensional system, rather than from a Hamiltonian function of finite-dimensional approximate system. Finally, the spatial discretization of these schemes is used in simulation. Compared with the known results by un-symplectic algorithms, the numerical solutions by the symplectic algorithm are stable in long-time simulation. Compared with the traditional symplectic algorithm for solving infinite-dimensional systems, our algorithm is more efficient and more accurate.
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