Hamilton系统下基于相位误差的精细辛算法
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摘要
Hamilton系统是一类重要的动力系统,辛算法(如生成函数法、SRK法、SPRK法、多步法等)是针对Hamilton系统所设计的具有保持相空间辛结构不变或保Hamilton函数不变的算法.但是,时域上,同阶的辛算法与Runge-Kutta法具有相同的数值精度,即辛算法在计算过程中也存在相位误差,导致时域上解的数值精度不高.经过长时间计算后,计算结果在时域上也会变得"面目全非".为了提高辛算法在时域上解的精度,将精细算法引入到辛差分格式中,提出了基于相位误差的精细辛算法(HPD-symplectic method),这种算法满足辛格式的要求,因此在离散过程中具有保Hamilton系统辛结构的优良特性.同时,由于精细化时间步长,极大地减小了辛算法的相位误差,大幅度提高了时域上解的数值精度,几乎可以达到计算机的精度,误差为O(10~(-13)).对于高低混频系统和刚性系统,常规的辛算法很难在较大的步长下同时实现对高低频精确仿真,精细辛算法通过精细计算时间步长,在大步长情况下,没有额外增加计算量,实现了高低混频的精确仿真.数值结果验证了此方法的有效性和可靠性.
Symplectic methods, including the generating function method, the symplectic Runge-Kutta(RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10~(-13). For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.
Symplectic methods, including the generating function method, the symplectic Runge-Kutta(RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10~(-13). For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.
引文
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[15] 钟万勰,吴峰,孙雁,等.保辛水波动力学[J].应用数学和力学,2018,39(8):855-874.(ZHONG Wanxie,WU Feng,SUN Yan,et al.Symplectic water wave dynamics[J].Applied Mathematics and Mechanics,2018,39(8):855-874.(in Chinese))
[16] 富明慧,李勇息.求解病态线性方程组的预处理精细积分法[J].应用数学和力学,2018,39(4):462-469.(FU Minghui,LI Yongxi.A preconditioned precise integration method for solving ill-conditioned linear equations[J].Applied Mathematics and Mechanics,2018,39(4):462-469.(in Chinese))
[17] HUANG Y A,DENG Z C,YAO L X.An improved symplectic precise integration method for analysis of the rotating rigid-flexible coupled system[J].Journal of Sound and Vibration,2007,299(1/2):229-246.
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[19] 黄永安,尹周平,邓子辰,等.多体动力学的几何积分方法研究进展[J].力学进展,2009,39(1):44-57.(HUANG Yongan,YIN Zhouping,DENG Zicheng,et al.Progress in geometric integration method for multibody dynamics[J].Advances in Mechanics,2009,39(1):44-57.(in Chinese))
[20] 徐明毅,张勇传.精细辛算法的高效格式和简化计算[J].力学与实践,2005,27(1):55-57.(XU Mingyi,ZHANG Yongchuan.Efficient format and simple computation of precise symplectic integration method[J].Mechanics in Engineering,2005,27(1):55-57.(in Chinese))
[21] BRUSA L,NIGRO L.A one-step method for direct integration of structural dynamic equations[J].International Journal for Numerical Methods in Engineering,1980,15(5):685-699.
[22] DAVID C,ERNST H,LUBICH C.Numerical energy conservation for multi-frequency oscillatory differential equations[J].BIT Numerical Mathematics,2005,45(2):287-305.
[2] G?RTZ P.Backward error analysis of symplectic integrators for linear separable Hamiltonian systems[J].Journal of Computational Mathematics,2002,20(5):449-460.
[3] 邢誉峰,杨蓉.单步辛算法的相位误差分析及修正[J].力学学报,2007,39(5):668-671.(XING Yufeng,YANG Rong.Phase errors and their correction in symplectic implicit single-step algorithm[J].Chinese Journal of Theoretical and Applied Mechanics,2007,39(5):668-671.(in Chinese))
[4] MONOVASILIS T,KALOGIRATOU Z,SIMOS T E.Symplectic partitioned Runge-Kutta methods with minimal phase-lag[J].Computer Physics Communications,2010,181(7):1251-1254.
[5] SIMOS T E.A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schr?dinger equation[J].Journal of Mathematical Chemistry,2011,49(10):2486-2518.
[6] MONOVASILIS T,KALOGIRATOU Z,SIMOS T E.Two new phase-fitted symplectic partitioned Runge-Kutta methods[J].International Journal of Modern Physics C,2011,22(12):1343-1355.
[7] 陈璐,王雨顺.保结构算法的相位误差分析及其修正[J].计算数学,2014,36(3):271-290.(CHEN Lu,WANG Yushun.Phase error analysis and correction of structure preserving algorithms[J].Mathematica Numerica Sinica,2014,36(3):271-290.(in Chinese))
[8] VYVER H V D.A symplectic Runge-Kutta-Nystrom method with minimal phase-lag[J].Physics Letters A,2007,367(1/2):16-24.
[9] MONOVASILIS Th,KALOGIRATOU Z,SIMOS T E.Exponentially fitted symplectic Runge-Kutta-Nystr?m methods[J].Applied Mathematics & Information Sciences,2013,7(1):81-85.
[10] 刘晓梅,周钢,王永泓,等.辛算法的纠飘研究[J].北京航空航天大学学报,2013,39(1):22-26.(LIU Xiaomei,ZHOU Gang,WANG Yonghong,et al.Rectifying drifts of symplectic algorithm[J].Journal of Beijing University of Aeronautics and Astronautics,2013,39(1):22-26.(in Chinese))
[11] MONOVASILIS Th.Symplectic partitioned Runge-Kutta methods with the phase-lag property[J].Applied Mathematics and Computation,2012,218(18):9075-9084.
[12] XI X P,SIMOS T E.A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schr?dinger equation and related problems[J].Journal of Mathematical Chemistry,2016,54(7):1417-1439.
[13] 朱帅,周钢,刘晓梅,等.精细辛有限元方法及其相位误差研究[J].力学学报,2016,48(2):399-405.(ZHU Shuai,ZHOU Gang,LIU Xiaomei,et al.Precise symplectic time finite element method and the study of phase error[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(2):399-405.(in Chinese))
[14] 钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报,1994,37(2):131-136.(ZHONG Wanxie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,37(2):131-136.(in Chinese))
[15] 钟万勰,吴峰,孙雁,等.保辛水波动力学[J].应用数学和力学,2018,39(8):855-874.(ZHONG Wanxie,WU Feng,SUN Yan,et al.Symplectic water wave dynamics[J].Applied Mathematics and Mechanics,2018,39(8):855-874.(in Chinese))
[16] 富明慧,李勇息.求解病态线性方程组的预处理精细积分法[J].应用数学和力学,2018,39(4):462-469.(FU Minghui,LI Yongxi.A preconditioned precise integration method for solving ill-conditioned linear equations[J].Applied Mathematics and Mechanics,2018,39(4):462-469.(in Chinese))
[17] HUANG Y A,DENG Z C,YAO L X.An improved symplectic precise integration method for analysis of the rotating rigid-flexible coupled system[J].Journal of Sound and Vibration,2007,299(1/2):229-246.
[18] 曾进,周钢.精细辛算法[J].上海交通大学学报,1997,31(9):31-33.(ZENG Jin,ZHOU Gang.Precise symplectic algorithm[J].Journal of Shanghai Jiaotong University,1997,31(9):31-33.(in Chinese))
[19] 黄永安,尹周平,邓子辰,等.多体动力学的几何积分方法研究进展[J].力学进展,2009,39(1):44-57.(HUANG Yongan,YIN Zhouping,DENG Zicheng,et al.Progress in geometric integration method for multibody dynamics[J].Advances in Mechanics,2009,39(1):44-57.(in Chinese))
[20] 徐明毅,张勇传.精细辛算法的高效格式和简化计算[J].力学与实践,2005,27(1):55-57.(XU Mingyi,ZHANG Yongchuan.Efficient format and simple computation of precise symplectic integration method[J].Mechanics in Engineering,2005,27(1):55-57.(in Chinese))
[21] BRUSA L,NIGRO L.A one-step method for direct integration of structural dynamic equations[J].International Journal for Numerical Methods in Engineering,1980,15(5):685-699.
[22] DAVID C,ERNST H,LUBICH C.Numerical energy conservation for multi-frequency oscillatory differential equations[J].BIT Numerical Mathematics,2005,45(2):287-305.