Multi-symplectic Runge-Kutta-type methods for Hamiltonian wave equations

IMA Journal of Numerical Analysis

Volumn 26, Issue 2, 2006, Pages 252-271

  Liu, Hongyu ^a,b   Zhang, Kai ^a,c  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^c JILIN UNIVERSITY   (China)

Author keywords

Hamiltonian PDEs; Multi symplectic; Non linear wave equations; Partitioned Runge Kutta method; Runge Kutta method; Stability analysis

Indexed keywords

HAMILTONIANS; WAVE EQUATIONS;

HAMILTONIAN PDES; MULTI-SYMPLECTIC; NONLINEAR WAVE EQUATION; PARTITIONED RUNGE-KUTTA METHOD; STABILITY ANALYSIS;

RUNGE KUTTA METHODS;

EID: 33645743266     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/dri042     Document Type: Article

References (26)

1. ABBOTT, M. B. & BASCO, D. R. (1989) Computational Fluid Dynamics. New York: John Wiley.
2. ASCHER, U. M. & MCLACHLAN, R. I. (2004) Multisymplectic box schemes and the Korteweg-de Vries equation. Appl. Numer. Math., 48, 255-269.
3. BRIDGES, T. J. (1997a) A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities. Proc. R. Soc. Lond. A, 453, 1365-1395.
4. BRIDGES, T. J. (1997b) Multi-symplectic structures and wave propagation. Math. Proc. Camb. Philos. Soc., 121, 147-190.
5. BRIDGES, T. J. & DERKS, G. (1999) Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry. Proc. R. Soc. Lond. A, 455, 2427-2469.
6. BRIDGES, T. J. & DERKS, G. (2002a) Linear stability of solitary wave solutions of the Kawahara equation and its generalizations. SIAM J. Math. Anal., 33, 1356-1378.
7. BRIDGES, T. J. & DERKS, G. (2002b) Stability of solitary waves of the fifth-order KdV equation: a numerical framework. Physica D, 172, 190-216.
8. BRIDGES, T. J. & REICH, S. (2001a) Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A, 284, 184-193.
9. BRIDGES, T. J. & REICH, S. (2001b) Multi-symplectic spectral discretization for the Zakharov-Kuznetsov and shallow-water equations. Physica D, 152, 491-504.
10. CANDY, J. & ROZMUS, W. (1991) A symplectic integration for separable Hamiltonian functions. J. Comput. Phys., 92, 230-256.
11. FOREST, F. & RUTH, R. D. (1990) Fourth-order symplectic integration. Physica D, 43, 105-117.
12. HAIRER, E., LUBICH, C. & ROCHE, M. (1989) The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Berlin: Springer.
13. HAIRER, E., LUBICH, C. & WANNER, G. (2002) Geometric Numerical Integration. Berlin: Springer.
14. HONG, J. L., LIU, H. Y. & SUN, G. (2005a) Spurious behavior of a symplectic integrator. Comput. Math. Appl. (to appear).
15. HONG, J. L., LIU, H. Y. & SUN, G. (2005b) The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs. Math. Comp. (Epub ahead of print September 29, 2005: to appear).
16. HONG, J. L. & LIU, Y. (2003) A novel numerical approach to simulating nonlinear Schrödinger equations with varying coefficients. Appl. Math. Lett., 16, 759-765.
17. LIU, H. Y. & SUN, G. (2004) Symplectic RK methods and symplectic PRK methods with real eigenvalues. J. Comput. Math., 22, 769-776.
18. MARSDEN, J. E., PATRICK, G. P. & SHKOLLER, S. (1999) Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys., 199, 351.
19. MARSDEN, J. E. & SHKOLLER, S. (1999) Multi symplectic geometry, covariant Hamiltonians and water waves. Math. Proc. Camb. Philos. Soc., 125, 553.
20. MEIS, T. & MARCOWITZ, U. (1981) Numerical Solution of Partial Diffrential Equations. New York, Berlin: Springer.
21. MOORE, B. E. (2003) A modified equations approach for multi-symplectic integration methods. Ph.D. Thesis, The University of Surrey, Guildford, UK.
22. MOORE, B. E. & REICH, S. (2003) Backward error analysis for multi-symplectic integration methods. Numer. Math., 95, 625-652.
23. REICH, S. (2000a) Finite volume methods for multi-symplectic PDEs. BIT, 40, 559-582.
24. REICH, S. (2000b) Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. J. Comp. Phys., 157, 473-499.
25. SANZ-SERNA, J. M. & CALVO, M. P. (1994) Numerical Hamiltonian Problems. London: Chapman and Hall.
26. ZHAO, P. F. & QIN, M. Z. (2000) Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation. J. Phys. A, 33, 3613-3626.