Symplectic geometry plays a very important role in the research and development of Hamilton mechanics, which has been attracting increasing interest. Consequently, the study of the numerical methods with symplectic nature becomes a necessity.
Feng Kang introduced in [5] the concept of symplectic scheme of the Hamilton equation, and used the generating function methods to construct the symplectic scheme with arbitrarily precise order in the finite dimensional case, which can be applied to the ordinary differential equation, such as the two body problem. He also widened the traditional concept of generating function.
The authors in this paper use the method in the infinite dimensional case following [6], that is, using generating function methods to construct the difference scheme of arbitrary order of accuracy for partial differential equations which can be written as Hamilton system in the Banach space.