变分与无限维系统的高精度辛格式

VARIATION AND HIGH ORDER ACCURACY SYMPLECTIC SCHEME FOR INFINITE DIMENSIONAL SYSTEM

1.引 言 冯康和他的研究小组提出的生成函数法[1]系统地解决了象二体问题这样地有限维Hamil-ton系统辛算法的构造问题,该方法也可以自然地推广到无限维Hamilton系统[2].首先在空间方向进行离散,例如采用差分或谱离散,得到有限维Hamilton系统,然后再采用生成函数法离散该系统.这样得到的辛格式是整个一层的格式,对于研究格式的局部性质如多辛性质[3],局部能量守恒性质[5]就相当困难.

In this paper, we present a new method to construct the symplectic schemes by the third type generating functions [1] for infinite dimension Hamilton system. After overcoming successfully the essential difficult on the calculus of high order variations, we get the semi-discretization with arbitrary order of accuracy in time direction for the PDEs. Furthermore the related modified equations of original equation are obtained from the semi-discretization. Numerical results on collision of solitons are also presented to show the effectiveness of the scheme.