基于MQ拟插值的Sine-Gordon方程自适应保辛数值解法  

Adaptive Conservative Numerical Method of Sine-Gordon Equation Based on Multi-quadric Quasi-interpolation

作　　者：高钦姣[1] 张胜刚[2,3] 朱春钢 谢风媛[1] Gao Qinjiao;Zhang Shenggang;Zhu Chungang;Xie Fengyuan(School of Business,Dalian University of Foreign Language,Dalian 116044,2Department of Health Statistics,Dalian Medical University;Dalian 116044,3School of Mathematical Sciences,Dalian University of Technolog;Dalian 11600)

机构地区：[1]大连外国语大学商学院,大连116044 [2]大连医科大学公共卫生学院,大连116044 [3]大连理工大学数学科学学院,大连116002

出　　处：《计算机辅助设计与图形学学报》2018年第7期1224-1229,共6页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家自然科学基金(61501082);辽宁教育厅青年项目(2017JYT07);大连外国语大学教学研究项目(2017Y30)

摘　　要：为了提高算法的长期跟踪能力和计算效率,利用MQ(multi-quadric)拟插值构造SG(Sine-Gordon)方程的一种自适应保辛数值算法.首先使用带有对称核函数的MQ拟插值拟合其潜在的函数及其导数值;然后根据节点移动方程移动节点位置得到下一时间层的节点组;最后在时间方向将SG方程使用Staggered St?rmer Verlet算法进行离散,得到新的节点组在下一时间层对应的数值解,文中给出了算法的能量保持误差估计以及截断误差估计.数值实验结果表明,该算法操作简便、精度高、具备长期跟踪能力.Existing numerical methods are weak in simulating the SG（Sine-Gordon） equation for long time. They may also be time-consuming and inefficient when the solutions involve large variations. This paper proposes an adaptive and energy conservative approach based on the MQ（multi-quadric） to overcome these limits. Firstly, the MQ quasi-interpolations with symmetric kernels are employed to approximate the spatial derivatives of each variable; secondly the new knots on the next time step are obtained according to the moving knots equation; thirdly the Staggered St？rmer Verlet scheme is employed to approximate the temporal derivatives of each variable. The energy conservation estimation and the truncation error of the proposed scheme are presented. Numerical experiments demonstrate that the proposed method is easy to implement, accurate and able to simulate the SG equation for long time.

关 键 词：SINE-GORDON方程 自适应算法 保辛算法 MQ拟插值 

分 类 号：O241.8[理学—计算数学]