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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

Symplectic weighted discontinuous Galerkin method with minimal phase-lag
ZHU Shuai1 , ZHOU Gang2 , LIU Xiaomei3 , WENG Shilie1
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; Accepted: 2015-11-13; Published online: 2015-12-17 14:41
Foundation item: National Natural Science Foundation of China (50876066); Shanghai University Youth Teacher Training Program (ZZZZEGD15007); Shanghai Second Polytechnic University Fund Project (EGD15XQD14); Shanghai Second Polytechnic University Applied Mathematics Key Discipline (XXKZD1304)
Corresponding author. WENG Shilie, E-mail:slweng@sjtu.edu.cn
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

1 Hamilton系统及相位误差

 (1)

 (2)

 (3)

 (4)

2 加权间断有限元方法

 (5)

 (6)

 (7)

 (8)
 (9)

 图 1 计算过程 Fig. 1 Computational procedure

 (10)

3 极小化相位误差

 (11)

 (12)

 图 2 ωh和最优权重αopt的关系 Fig. 2 Relationship between ωh and αopt

A是辛矩阵。

 (13)

θ=θ′，所以TαTα_sym具有相同的相位误差。即对极小化相位误差的矩阵Tα进行式(13)的变换，得到新的Jacobi矩阵Tα_sym同样具有极小的相位误差。

4 数值算例 4.1 椭圆型Hamilton系统

 图 3 相图 Fig. 3 Phase portrait

 参数 误差绝对值的最大值 WDG-PF TFE2 [15] AVF [21] p(t) 9.7382×10 -15 2.3999 2.3660 q(t) 9.6202×10 -17 0.1461 1.7997 能量 3.3061×10 -38 1.3767×10 -14 0

 参数 极小相位误差 WDG-PF FSJS[13] SPRK[10] RKN[8] p(t) 9.7382×10-15 0.1859 0.0072 0.8988 q(t) 9.6202×10-17 0.1084 0.0121 0.0030 能量 3.3061×10-38 0.0549 0.0263 1.3493

 图 4 一次加权间断元方法在前4个单元内部误差图(h=0.1) Fig. 4 Error distribution of one-order WDG-PF in first four elements(h=0.1)

4.2 高低混频Hamilton系统

Hamilton函数：

 图 5 几种不同方法求解p1、q1、p2和q2误差 Fig. 5 Errors of p1, q1, p2 and q2 solved by different methods

 方法 能量(误差) t=200 s t=400 s t=600 s WDG-PF 1.919×10-12 4.019×10-12 5.983×10-12 TFE1[15] 9.164×10-11 1.213×10-10 2.7×10-10 FSJS[13] 0.143 0.019 44 0.085 56 SPRK[10] 6.998×10-5 4.388×10-5 4.294×10-5

 图 6 4次WDG-PF在前4个单元内部误差情况 Fig. 6 Error distribution for four-order WDG-PF in first four elements
 图 7 3次WDG-PF在前4个单元内部误差情况 Fig. 7 Error distribution for third-order WDG-PF in first four elements

 图 8 4次WDG-PF在前4个单元内部导数误差情况 Fig. 8 Derivative error distribution for four-order WDG-PF in first four elements
 图 9 3次WDG-PF在前4个单元内部导数误差情况 Fig. 9 Derivative error distribution for third-order WDG-PF in first four elements

5 结论

1) 对于给定的Hamilton系统和单元长度，使得相位误差极小且保辛，权重α“一次计算，终身使用”。

2) 可以解决间断有限元方法不能保证Hamilton系统辛结构的缺点，具有极小的相位误差和极高精度的保能量特性，几乎达到计算机精度。

3) 这种加权间断有限单元有超收敛点。

4) 特别针对高低混频Hamilton系统或者刚性问题，WDG-PF方法可以在大步长下同时实现对高频和低频信号的准确仿真。

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#### 文章信息

ZHU Shuai, ZHOU Gang, LIU Xiaomei, WENG Shilie

Symplectic weighted discontinuous Galerkin method with minimal phase-lag

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(8): 1682-1690
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0523