辛差分格式的阶条件和高阶格式的构造

Order Conditions for Symplectic Schemes and Constructions of High order Schemes

哈密尔顿系统具有很多本质的性质，如保面积、保能量和动量等，而最重要的是自治哈密尔顿系统相流的辛性质，这为保相面积和相体积提供了保证，因此我们希望差分格式在被用来求哈密尔顿系统的数值解时能保持这种辛性质。在文献［１─３］中，冯康等人发展了哈密尔顿系统辛差分格式的一整套理论。本文主要考虑辛ＲＫＮ方法及显式辛格式的阶条件，并给出一些高阶辛ＲＫＮ格式，还提出一种利用低阶格式构造高阶格式的方法。此方法对辛格式和非辛格式均适用。限于篇幅，本文略去大部分定理的证明并尽可能简化说明。

in[1-3], Feng Kang and his colleagues developed a systematic research of canonical difference schemes for Hamiltonian systems. In this paper,order conditions for symplectic RKN methods are simplified under the symplectic conditions and the order conditions for explicit symplectic schemes are derived and simplified. We will also introduce the concept of adjoint methods and some properties of them to show that there is a self-adjoint scheme of even order corresponding to every method. Using the self-adjoint schemes with lower order,we can construct higher order schemes by multiplicative extrapolation method,and this constructing process can be continued to get arbitrary even order schemes. The multiplicative extrapolation method presented here can be used to non-symplectic systems as well as symplectic ones.