Splitting Methods for Non-autonomous Hamiltonian Equations

Journal of Computational Physics

Volumn 170, Issue 1, 2001, Pages 205-230

  Blanes, S ^a   Moan, P C ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Free Lie algebra; Initial value problems; Magnus expansion; Non linear time dependent differential equations; Numerical methods; Symplectic

Indexed keywords

ALGEBRA; HAMILTONIANS; INITIAL VALUE PROBLEMS;

FREE LIE ALGEBRA; INITIAL-VALUE PROBLEM; LIE ALGEBRA; LINEAR TIME; MAGNUS EXPANSION; NON LINEAR; NON-LINEAR TIME-DEPENDENT DIFFERENTIAL EQUATION; NONAUTONOMOUS; SYMPLECTIC; TIME DEPENDENT;

NUMERICAL METHODS;

EID: 0001648558     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.2001.6733     Document Type: Article

References (30)

1. Arnold V. I. Mathematical Methods of Classica-Mechanics. 1989.
2. Blanes S., Casas F., Oteo J. A., Ros J. Magnus and Fer expansions for matrix differential equations: The convergence problem. J. Phys. A Math. Gen. 31:1998;259.
3. Blanes S., Cases F., Ros J. Symplectic integrators with processing: A general study. SIAM J. Sci. Comput. 21:1999;711.
4. Blanes S., Casas F., Ros J. Processing symplectic methods for near-integrable Hamiltonian systems. Celest. Mech. Dyn. Astron. 70:2000;17.
5. Blanes S., Casas F., Ros J. Improved high order integrators based on Magnus expansion. BIT. 40:2000;434.
6. Blanes S., Moan P. C. Splitting methods for the time-dependent Schrödinger equation. Phys Lett. A. 265:2000;35.
7. Blanes S., Moan P. C. Practical Symplectic Partitioned Runge-Kutta and Runge-Kutta-Nyström Methods. 2000.
8. Candy J., Rozmus W. A symplectic integration algorithm for separable Hamiltonian functions. J. Comput. Phys. 92:1991;230.
9. Cary J. R., Doxas J. An explicit sympletic integration scheme for plasma simulations. J. Comput. Phys. 107:1993;98.
10. Castell F., Gaines J. An efficient approximation method for stochastic differential equations by means of the exponential Lie series. Math. Comput. Simulation. 38:1995;1.
11. A. J. Dragt, Lectures on nonlinear orbit dynamics, in Physics of High Energy Particle Accelerators, AIP Conf. Proc., edited by R. A. Carrigan, F. R. Huson, and M. MonthAm. Inst. of Phys. New York, 1982, Vol. 87, p. 147.
12. Forest E. Sixth-order Lie group integrators. J. Comput. Phys. 99:1992;209.
13. Gray S., Manolopoulos D. E. Symplectic integrators tailored to the time-dependent Schrödinger equation. J. Chem. Phys. 104:1996;7099.
14. Iserles A., Nørsett S. P. On the solution of linear differential equations in Lie groups. Philos. Trans. R. Soc. A. 357:1999;983.
15. P.-V. Koseleff, Relations among Lie formal series and construction of symplectic integrators, in, Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, AAECC-10, edited by, G. Cohen, T. Mora, and O. Moreno, Springer-Verlag, New York, 1993.
16. Leoforestier C., Bisseling R. H., Cerjam C., Feit M. D., Friesner R., Guldeberg A., Hammerich A., Jolicard G., Karrlein W., Meyer H.-D., Plipkin N., Roncero O., Kosloff R. A comparison of different propagation schemes for the time dependent Schroedinger equation. J. Comput. Phys. 94:1991;59.
17. McLachlan R. I., Atela P. The accuracy of symplectic Integrators. Nonlnearity. 5:1992;541.
18. McLachlan R. I. Composition methods in the presence of small parameters. BIT. 35:1995;258.
19. McLachlan R. I. On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16:1995;151.
20. McLachlan R. I., Gray S. K. Optimal stability polynomials for spliting methods, with application to the time-dependent Schrödinger equation. Appl. Numer. Math. 25:1997;275.
21. Magnus W. On the exponential solution of differential equations for a linear operator. Commun Pure Appl. Math. 7:1954;649.
22. Moan P. C. Efficient Approximation of Sturm-Liouville Problem Using Lie-Group Methods. 1998.
23. Moan P. C., Oteo J. A., Ros J. On the existence of the exponential solution of linear differential systems. J. Phys. A. Math. Gen. 32:1999;5133.
24. Oteo J. A., Ros J. The Magnus expansion for classical Hamiltonian system. J. Phys. A Math. Gen. 24:1991;5751.
25. Raj S. Paul, Rajasekar S., Murali K. Coexisting chaotic attractors, their basin of attraction and synchronization of chaos in two coupled Duffing oscillators. Phys. Lett. A. 264:1999;283.
26. Ruth R. A canonical integration technique. IEEE Trans. Nucl. Sci. 30:1983;2669.
27. Sanz-Serna J. M., Calvo M. P. Numerical Hamiltonian Problems. 1994.
28. Suzuki M. Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A. 146:1990;319.
29. Wisdom J., Holman M. N-body problem, Astron. J. 102:1991;1528.
30. Yoshida H. Construction of higher order symplectic integrators. Phys. Lett. A. 150:1990;262.