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Runge-kutta method, finite element method, and regular algorithms for hamiltonian system

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Abstract

The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic RungeKutta (RK) method is an important part of the former, and the continuous finite element method (CFEM) belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms—the regular method. Finally, numerical experiments are given to verify the theoretical results.

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References

  1. Arnold, V. I. Mathematical Methods of Classical Mechanics, Springer, New York (1978)

    MATH  Google Scholar 

  2. Ge, Z. and Marsden, J. E. Lie-Possion Hamilton-Jacobi theory and Lie-Poisson intergrators. Phys. Lett. A, 133(3), 134–139 (1988)

    Article  MathSciNet  Google Scholar 

  3. Feng, K. On difference schemes and symplectic geometry. Proceedings of Symposium Differential Geometry and Differential Equations-Computation of Differential Equations, Science Press, Beijing, 42–58 (1985)

    Google Scholar 

  4. Feng, K. Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math., 4, 279–289 (1986)

    MathSciNet  MATH  Google Scholar 

  5. Feng, K. How to compute properly Newton’s equation of motion. Proceedings of 2nd Conference on Numerical Methods for Partial Differential Equations, World Scientific, Singapore, 15–22 (1992)

    Google Scholar 

  6. Feng, K. and Qin, M. Z. Symplectic Geometric Algorithms for Hamiltonian Systems (in Chinese), Zhejiang Science and Technology Press, Hangzhou (2003)

    Google Scholar 

  7. Sanz-Serna, J. M. Runge-Kutta schemes for Hamiltonian systems. BIT Numerical Mathematics, 28, 877–883 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sanz-Serna, J. M. and Calvo, P. M. Numerical Hamiltonian Problems, Chamman Hall, London (1994)

    MATH  Google Scholar 

  9. Lasagni, F. M. Canonical Runge-Kutta methods. Z. Angew. Math. Phys., 39, 952–953 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Suris, Y. B. On the preservation of the symplectic structure in the course of numerical integration of Hamiltonian systems (in Russian). Numerical Solution of Ordinary Differential Equations, USSR Academy of Sciences, Moscow, 148–160 (1988)

    Google Scholar 

  11. Sun, G. Symplectic partitioned Runge-Kutta methods. J. Comput. Math., 11, 365–372 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Sun, G. Construction of high order symplectic Runge-Kutta methods. J. Comput. Math., 13, 40–50 (1995)

    MathSciNet  MATH  Google Scholar 

  13. Tang, Y. F. The symplecticity of multi-step methods. Comput. Math. Appl., 25, 83–90 (1993)

    Article  MATH  Google Scholar 

  14. Tang, Y. F. Symplectic computation of Hamiltonian systems (I). J. Comp. Math., 20, 267–276 (2002)

    MATH  Google Scholar 

  15. Hairer, E. Conjugate-symplecticity of linear multistep methods. J. Comp. Math., 26, 657–659 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Bridges, T. J. and Reich, S. Numerical methods for Hamiltonian PDEs. J. Phys. A: Math. Gen., 39, 5287–5320 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhong, W. X. Time domain FEM and symplectic conservation (in Chinese). Journal of Mechanical Strength, 27(2), 178–183 (2005)

    Google Scholar 

  18. Chen, C. M. and Tang, Q. Study of finite elements for nonlinear Hamiltonian systems. The Fifth International Conference on PDE’s and Numerical Analysis, Changsha (2006)

    Google Scholar 

  19. Tang, Q. and Chen, C. M. Energy conservation and symplectic properties of continuous finite element methods for Hamiltonian systems. Appl. Math. Comput., 181, 1357–1368 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tang, Q., Chen, C. M., and Liu, L. H. Continuous finite element methods for Hamiltonian systems. Appl. Math. Mech.-Engl. Ed., 28(8), 1071–1080 (2007) DOI 10.1007/s10483-007-0809-y

    Article  MATH  Google Scholar 

  21. Tang, Q. and Chen, C. M. Space-time finite element method for the Schrodinger equation and its conservation. Appl. Math. Mech.-Engl. Ed., 27(3), 335–340 (2006) DOI 10.1007/s10483-006-0308-z

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, C. M., Tang, Q., and Hu, S. F. Finite element method with superconvergence for nonlinear Hamiltonian systems. J. Comp. Math., 29, 167–184 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Quispel, G. R. W. and McLaren, D. I. A new class of energy-preserving numerical integration methods. J. Phys. A: Math. Theor., 41, 045206 (2008)

    Article  MathSciNet  Google Scholar 

  24. Hairer, E. Energy-preserving variant of collocation methods. Journal of Numerical Analysis, Industrial and Applied Mathematics, 5(1–2), 73–84 (2010)

    MathSciNet  Google Scholar 

  25. Hairer, E., Lubich, C., and Wanner, G. Geometric Numerical Integration-Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin (2002)

    MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, P. R. China

    Shu-fang Hu  (胡姝芳) & Chuan-miao Chen  (陈传淼)

  2. Institute of Mathematics and Physics, Central South University of Forestry and Technology, Changsha, 410004, P. R. China

    Shu-fang Hu  (胡姝芳)

Authors
  1. Shu-fang Hu  (胡姝芳)
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  2. Chuan-miao Chen  (陈传淼)
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Corresponding author

Correspondence to Shu-fang Hu  (胡姝芳).

Additional information

Project supported by the National Natural Science Foundation of China (No. 11071067) and the Hunan Graduate Student Science and Technology Innovation Project (No.CX2011B184)

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Hu, Sf., Chen, Cm. Runge-kutta method, finite element method, and regular algorithms for hamiltonian system. Appl. Math. Mech.-Engl. Ed. 34, 747–760 (2013). https://doi.org/10.1007/s10483-013-1704-8

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  • DOI: https://doi.org/10.1007/s10483-013-1704-8

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2010 Mathematics Subject Classification

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