The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

Mathematics of Computation

Volumn 75, Issue 253, 2005, Pages 167-181

  Hong, Jialin ^a   Liu, Hongyu ^b   Sun, Geng ^b  

^a INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

^b CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Hamiltonian partial differential equation; Multi symplecticity; Partitioned Runge Kutta method

Indexed keywords

EID: 85009774577     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-05-01793-X     Document Type: Article

References (16)

1. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987. MR0878564 (88d:65002)
2. Th. J. Bridges and S. Reich, Multi-symplectic integrator: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Physics Letters A, 284(4-5) (2001), 184-193. MR1854689 (2002g:65166)
3. G.J. Cooper, Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal., 7 (1987), 1-13. MR0967831 (90d:65133)
4. D.B. Duncan, Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 34 (1997), 1742-1760. MR1472194 (98m:65139)
5. J. de Frutos & J.M. Sanz-Serna, Split-step spectral schemes for nonlinear Dirac systems, J. Comput. Phys., 83 (1989), 407-423. MR1013060 (91f:81021)
6. P. Görtz and R. Scherer, Hamiltonian systems and symplectic integrators, Nonlinear Analysis TMA, 30 (1997), 1887-1892. MR1490098 (98j:65049)
7. A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge, 1996. MR1384977 (97m:65003)
8. J.E. Marsden & M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 1-158. MR2009697 (2004h:37130)
9. S. Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000) 2, 473-499. MR1739109 (2001a:65087)
10. J.M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (1988), 877-883. MR0972812 (90b:65145)
11. J.M. Sanz-Serna, The numerical integration of Hamiltonian systems. In: Computational Ordinary Differential Equations, ed. by J.R. Cash & I.Gladwell, Clarendon Press, Oxford, (1992) 437-449. MR1387155
12. J.M. Sanz-Serna and M.P. Calva, Numerical Hamiltonian Systems, Chapman & Hall, London, 1994. MR1270017 (95f:65006)
13. G. Sun, A simple way of constructing symplectic Runge-Kutta methods, J. Comput. Math., 18 (2000), 61-68. MR1741173 (2000j:37118)
14. G. Sun, Symplectic partitioned Runge-Kutta methods, J. Comput. Math., 11 (1993), 365-372. MR1252773 (95a:65113)
15. Y.B. Suris, On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems (in Russian), In: Numerical Solution of Ordinary Differential Equations, ed. S.S. Filippov, Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow, 1988, 148-160.
16. Y.B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpolation (in Russian), Math. Model., 2 (1990), 78-87. MR1064467 (92b:65051)