Multi-symplectic integration methods for Hamiltonian PDEs

Future Generation Computer Systems

Volumn 19, Issue 3, 2003, Pages 395-402

  Moore, Brian E ^a   Reich, Sebastian ^b  

^a UNIVERSITY OF SURREY   (United Kingdom)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Backward error analysis; Conservation laws; Modified equations; Multi symplectic PDEs; Preissman box scheme

Indexed keywords

APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; ERROR ANALYSIS; INTEGRATION;

CONSERVATION LAWS;

HAMILTONIANS;

EID: 0037400145     PISSN: 0167739X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-739X(02)00166-8     Document Type: Article

References (21)

1. M.B. Abbott, Computational Hydraulics, Pitman, London, 1979.
2. M.B. Abbott, D.R. Basco, Computational Fluid Dynamics, Longman Scientific, Harlow, 1989.
3. 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;1117-1143.
4. Bridges T.J., Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc. 121:1997;147-190.
5. Bridges T.J., A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities. Proc. Roy. Soc. London Ser. A. 453:1997;1365-1395.
6. Bridges T.J., Derks G., Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry. Proc. Roy. Soc. London Ser. A. 455:1999;2427-2469.
7. Bridges T.J., Derks G., The symplectic Evans matrix, and the instability of solitary waves and fronts with symmetry. Arch. Rat. Mech. Anal. 156:2001;1-87.
8. Bridges T.J., Reich S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A. 284:2001;184-193.
9. De Frutos J., Sanz-Serna J.M., Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation. Numer. Math. 75:1997;421-445.
10. P. Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich, R.D. Skeel (Eds.), Computational Molecular Dynamics: Challenges, Methods, Ideas, Lecture Notes in Computational Science and Engineering, vol. 4, Springer, Heidelberg, 1999.
11. P.G. Drazin, Solitons, Lecture Note Series 85, London Mathematical Society, Cambridge University Press, Cambridge, 1983.
12. Durán A., Sanz-Serna J.M., The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation. IMA J. Numer. Anal. 20:2000;235-261.
13. Hairer E., Lubich C., On the life-span of backward error analysis. Numer. Math. 76:1997;441-462.
14. Marsden J.E., Patrick G.P., Shkoller S., Multi-symplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199:1998;351-395.
15. B.E. Moore, A modified equations approach for multi-symplectic integration methods, Ph.D. Thesis, University of Surrey, in preparation.
16. B.E. Moore, S. Reich, Backward error analysis for multi-symplectic integration methods, submitted for publication.
17. P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
18. Reich S., Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36:1999;1549-1570.
19. Reich S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. J. Comput. Phys. 157:2000;473-499.
20. J.M. Sanz-Serna, M.P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, London, 1994.
21. G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974.