Landau-Ginzburg-Higgs方程的多辛Runge-Kutta方法

西北工业大学力学与土木建筑学院,西安 710072;2.西北工业大学动力与能源学院,西安 710072;3.大连理工大学工业装备结构分析国家重点实验室,辽宁大连 116023

基金项目: 国家自然科学基金资助项目(10772147;10632030);教育部博士点基金资助项目(20070699028);陕西省自然科学基金资助项目(2006A07);大连理工大学工业装备结构分析国家重点实验室开放基金资助项目(GZ0802)

Multi-Symplectic Runge-Kutta Methods for Landau-Ginzburg-Higgs Equation

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnincal University, Xi'an 710072, P. R. China;

摘要: 非线性波动方程作为一类重要的数学物理方程吸引着众多的研究者，基于Hamilton空间体系的多辛理论研究了Landau-Ginzburg-Higgs方程的多辛算法，讨论了利用Runge-Kutta方法构造离散多辛格式的途径，并构造了一种典型的半隐式的多辛格式，该格式满足多辛守恒律、局部能量守恒律和局部动量守恒律．数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性．
- 多辛 /
- Landau-Ginzburg-Higgs方程 /
- Runge-Kutta方法 /
- 守恒律 /
- 孤子解
Abstract: The nonlinear wave equation, describing many important physical phenomena, has been investigated widely in last several decades. Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, was sdudied based on the multisymplectic theory in Hamilton space. The multi symplectic Runge-Kutta method was reviewed and a semiimplicit scheme with certain discrete conservation laws was constructed to solve the first order partial differential equations that were derived from the LandauGinzburg-Higgs equation. The results of numerical experiment for soliton solution of the Landau-Ginzburg-Higgs equation were reported finally, which show that the multi symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.
- multi- symplectic /
- Landau-Ginzburg-Higgs equation /
- Runge-Kutta method /
- conservation law /
- soliton solution

[1]	Bridges T J.Multi-symplectic structures and wave propagation[J].Math Proc Camb Philos Soc，1997,121(1):147-190. doi: 10.1017/S0305004196001429
[2]	Moore B E,Reich S.Multi-symplectic integration methods for Hamiltonian PDEs[J].Future Generation Computer Systems,2003,19(3):395-402. doi: 10.1016/S0167-739X(02)00166-8
[3]	Bridges T J,Reich S.Multi-symplectic integrators:numerical schemes for Hamiltonian PDEs that conserve symplecticity[J].Phys Lett A,2001,284(4/5):184-193. doi: 10.1016/S0375-9601(01)00294-8
[4]	Reich S.Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations[J].Journal of Computational Physics,2000,157(2):473-499. doi: 10.1006/jcph.1999.6372
[5]	胡伟鹏，邓子辰，李文成.膜自由振动的多辛方法[J].应用数学和力学，2007,28(9):1054-1062.
[6]	胡伟鹏，邓子辰.广义Boussinesq方程的多辛方法[J].应用数学和力学,2008,29(7):839-845.
[7]	Benettin G,Giorgilli A.On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms[J].J Stat Phys,1994,74(5/6):1117-1143. doi: 10.1007/BF02188219
[8]	QIN Meng-zhao,ZHANG Mei-qing.Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations[J].Computers & Mathematics With Applications,1990,19(10):51-62.
[9]	莫嘉琪，王辉，林一骅.广义Landau-Ginzburg-Higgs方程孤子解的扰动理论[J].物理学报，2005,54(12):5581-5584.