Multi-Symplectic Runge-Kutta Collocation Methods for Hamiltonian Wave Equations

Journal of Computational Physics

Volumn 157, Issue 2, 2000, Pages 473-499

  Reich, Sebastian ^a  

^a UNIVERSITY OF SURREY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

GAUSSIAN DISTRIBUTION; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; RUNGE KUTTA METHODS; WAVE EQUATIONS;

COLLOCATION METHOD; CONSERVATION LAW; GAUSS-LEGENDRE; GENERALISATION; LEGENDRE COLLOCATIONS; MULTI-SYMPLECTIC; SPACE AND TIME; SYMPLECTIC; SYMPLECTIC CONSERVATION; SYMPLECTIC INTEGRATORS;

HAMILTONIANS;

EID: 0034687898     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1999.6372     Document Type: Article

References (21)

1. Abbott M. B., Basco D. R. Computational Fluid Dynamics. 1989.
2. Benettin G., Giorgilli A. On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74:1994;1117.
3. Bridges Th. J. Multi-symplectic structures and wave propagation. Math. Proc. Cambridge Philos. Soc. 121:1997;147.
4. Bridges Th. J. A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities. Proc. R. Soc. London A. 453:1997;1365.
5. Bridges Th. J., Derks G. Unstable eigenvalues, and the linearization about solitary waves and fronts with symmetry. Proc. R. Soc. London A. 455:1999;2427.
6. Th. J. Bridges, and, S. Reich, Multi-symplectic Integrators: Numerical Schemes for Hamiltonian PDEs That Conserve Symplecticity, Technical Report.
7. Cooper G. J. Stability of Runge-Kutta methods for trajectory problems. IMA J. Numer. Anal. 7:1987;1.
8. Duncan D. B. Symplectic finite difference approximations of the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal. 34:1997;1742.
9. Fornberg B. A Practical Guide to Pseudospectral Methods. 1998.
10. Jiménez S. Derivation of the discrete conservation laws for a family of finite difference schemes. Appl. Math. Comput. 64:1994;13.
11. Marsden J. E., Patrick G. P., Shkoller S. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199:1999;351.
12. Marsden J. E., Shkoller S. Multisymplectic geometry, covariant Hamiltonians and water waves. Math. Proc. Cambridge Philos. Soc. 125:1999;553.
13. McLachlan R. I. Symplectic integration of Hamiltonian wave equations. Numer. Math. 66:1994;465.
14. McLachlan R. I. On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16:1995;151.
15. Li S., Vu-Quoc L. Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal. 32:1995;1839.
16. Reich S. Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36:1999;1549.
17. S. Reich, Finite Volume Methods for Multi-Symplectic PDEs, BIT, in press.
18. Sanz-Serna J. M., Calvo M. P. Numerical Hamiltonian Systems. 1994.
19. Strauss W., Vázquez L. Numerical solution of a nonlinear Klein-Gordon equation. J. Comput. Phys. 28:1978;271.
20. Veselov A. P. Integrable discrete-time systems and difference operators. Funktsional. Anal. Prilozhen. 22:1988;1.
21. Vu-Quoc L., Li S. Invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation. Comput. Methods Appl. Mech. Eng. 107:1993;341.