北京航空航天大学学报 ›› 2016, Vol. 42 ›› Issue (8): 1682-1690.doi: 10.13700/j.bh.1001-5965.2015.0523

• 论文 • 上一篇    下一篇

极小化相位误差加权间断有限元辛方法

朱帅1, 周钢2, 刘晓梅3, 翁史烈1   

  1. 1. 上海交通大学 机械与动力工程学院, 上海 200240;
    2. 上海交通大学 数学系, 上海 200240;
    3. 上海第二工业大学 理学院, 上海 201209
  • 收稿日期:2015-08-10 出版日期:2016-08-20 发布日期:2016-02-18
  • 通讯作者: 翁史烈,E-mail:slweng@sjtu.edu.cn E-mail:slweng@sjtu.edu.cn
  • 作者简介:朱帅,男,博士研究生。主要研究方向:计算流体力学、结构力学、辛方法、有限元方法。E-mail:zhushuaisjtu@qq.com,zhushuai@sjtu.edu.cn;翁史烈,男,博士,院士,博士生导师。主要研究方向:燃气轮机总体设计、燃气轮机动态控制。E-mail:slweng@sjtu.edu.cn
  • 基金资助:
    国家自然科学基金(50876066);上海高校青年教师培养资助计划(ZZZZEGD15007);上海第二工业大学校基金(EGD15XQD14);上海第二工业大学应用数学重点学科(XXKZD1304)

Symplectic weighted discontinuous Galerkin method with minimal phase-lag

ZHU Shuai1, ZHOU Gang2, LIU Xiaomei3, WENG Shilie1   

  1. 1. School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
    2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China;
    3. Department of Mathematics, Shanghai Second Polytechnic University, Shanghai 201209, China
  • Received:2015-08-10 Online:2016-08-20 Published:2016-02-18

摘要: 对于线性Hamilton系统,辛差分方法可以保持系统的辛结构,有限元方法可以保证系统的辛性质并具有能量守恒特性。但辛差分方法和有限元方法时域上仍然存在相位误差,使得计算的精度不是很理想。提出极小化相位误差加权间断有限元辛方法(WDG-PF),该方法是辛方法,同时,对Hamilton系统的求解具有极小的相位误差。数值显示该方法可以保证Hamilton系统的能量守恒性。WDG-PF方法解决了时间有限元方法(TFE)存在的相位漂移现象,同时指出间断有限元方法可以通过加权处理达到保辛要求。WDG-PF方法相较于针对相位误差设计的计算格式分数步对称辛算法(FSJS)、辛Runge-Kutta-Nystrom(RKN)格式以及辛分块Runge-Kutta(SPRK)等方法,WDG-PF显著地减少相位误差,和显著提高Hamilton系统能量精度的优点。相位误差和能量误差几乎达到计算机精度。同时单元内部具有超收敛现象。特别针对高低混频Hamilton系统,传统方法很难在固定的步长下同时实现对高频和低频信号的精确仿真,WDG-PF方法则可以在大步长下同时实现对低频信号和高频信号的高精度仿真。数值显示,WDG-PF方法切实有效。

关键词: Hamilton系统, 间断有限元方法, 相位误差, 辛算法, 保能量

Abstract: Symplectic finite difference method (FDM) can keep the symplectic structure, and finite element method (FEM) can keep the symplectic structure as well as energy conservation for linear Hamiltonian systems. However, symplectic FDM and FEM still have phase errors for the numerical solution, so, the computational accuracy is not very well in time domain analysis. Symplectic weighted discontinuous Galerkin method with minimal phase-lag (WDG-PF) is proposed for Hamiltonian systems. This method is symplectic and can highly decrease the phase error, compared to traditional method for Hamiltonian systems. Meanwhile, WDG-PF can keep the conservation of energy as well as the symplectic structure of Hamiltonian systems. WDG-PF can solve the phase-lag problem of continuous Galerkin method, and WDG is symplectic by the technique of weight. Compared to symmetric symplectic(FSJS) algorithm, Runge-Kutta-Nystrom(SRKN) and symplectic partitioned Runge-Kutta (SPRK) methods which are aimed at increasing the accuracy of phase error, WDG-PF ismuch more accurate and increase the energy accuracy of Hamiltonian systems, tremedously. The phase error and Hamiltonian function error almost achieve the accuracy of computer. WDG-PF has the ultraconvergence point in each element. Especially, for the systems with high and low frequency signals, and seldom has a method can simulate the high and low frequency signals with a fixed time step, WDG-PF can effectively simulate the high and low frequency signals with large time step. The numerical experiments show its validity.

Key words: Hamiltonian systems, discontinuous Galerkin method, phase error, symplectic algorithm, energy-preserving

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