High Order Symplectic Schemes for the Sine-Gordon Equation
High Order Symplectic Schemes for the Sine-Gordon Equation
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DOI：
10.1143/jpsj.72.2731
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发表时间：
2003-11
影响因子：
1.7
通讯作者：
Yushun Wang;Bin Wang;Zhongzhen Ji;M. Qin
中科院分区：
文献类型：
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作者：
Yushun Wang;Bin Wang;Zhongzhen Ji;M. Qin
本文以Sine-Gordon方程为例，给出了一种构造Hamilton偏微分方程组辛格式的新方法。与以往处理偏微分方程组的辛方法不同，我们的方法是将偏微分方程组看作是Banach空间中的哈密顿系统，然后将母函数方法应用于哈密顿系统。在成功地克服了高阶变分导数计算的基本困难之后，我们得到了时间方向上具有任意精度阶的偏微分方程组的半离散差分格式。进一步，通过半离散得到了相应的无限维哈密顿系统的修正方程。我们使用中心差分算子来离散空间中的导数。由此得到的全离散辛格式可以达到任意阶精度。文中还给出了孤子碰撞的数值结果，证明了该格式的有效性。
In this paper, taking the sine-Gordon equation as an example, we present a new method to construct the symplectic schemes for Hamilton PDEs. Different from the previous symplectic methods dealing with PDEs, our method is that to view the PDEs as a Hamilton system in Banach space, then to apply the generating functions method to the Hamilton system. After overcoming successfully the essential difficulties on the calculations of high order variation derivatives, we get the semi-discrete difference schemes for the PDEs with arbitrary order of accuracy in time direction. Furthermore the corresponding modified equations of the infinite dimensional Hamiltonian system are obtained from the semi-discretization. We use the central difference operators to discretize the derivatives in space. Thus the resulting full discrete symplectic schemes can be of any order accuracy. Numerical results on collisions of solitons are also presented to show the effectiveness of the schemes.