Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation

Journal of Mathematical Chemistry

Volumn 40, Issue 3, 2006, Pages 257-267

  Monovasilis, Th ^a,b   Kalogiratou, Z ^a   Simos, T E ^b,c  

^a UNIVERSITY OF WESTERN MACEDONIA   (Greece)

^b UNIVERSITY OF PELOPONNESE   (Greece)

^c NONE   (Greece)

Author keywords

Eigenvalue problem; Exponentially and trigonometrically fitted; Schr dinger equation; Symplectic methods

Indexed keywords

EID: 33750286640     PISSN: 02599791     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10910-006-9167-9     Document Type: Conference Paper

References (38)

1. T.E. Simos, Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems, chemical modelling: application and theory, R. Soc. Chem. 2 (2002) 170-270
2. G. Vanden Berghe, M. Van Daele and H. Vande Vyver, Exponentially fitted algorithms: fixed or frequency dependent knot points?, Appl. Num. Anal. Comp. Math. 1(1)(2004) 49-65.
3. Th. Monovasilis, Z. Kalogiratou and T.E. Simos, Numerical solution of the two-dimensional time independent Schrödinger equation by symplectic schemes, Appl. Num. Anal. Comp. Math. 1(1)(2004) 195-204.
4. G. Psihoyios and T.E. Simos, Effective numerical approximation of Schrödinger type equations through multiderivative exponentially-fitted schemes, Appl. Num. Anal. Comp. Math. 1(1) (2004) 205-215
5. G. Psihoyios and T.E. Simos, Efficient numerical solution of orbital problems with the use of symmetric four-step trigonometrically fitted methods, Appl. Num. Anal. Comp. Math. 1(1)(2004) 216-222.
6. M. Van Daele and G. Vanden Berghe, Extended one-step methods: an exponential fitting approach, Appl. Num. Anal. Comp. Math. 1(2)(2004) 353-362.
7. D.S. Vlachos and T.E. Simos, Partitioned linear multistep method for long term integration of the N-body problem, Appl. Num. Anal Comp. Math. 1(2)(2004) 540-546
8. Th. Monovasilis and Z. Kalogiratou, Trigonometrically and exponentially fitted symplectic methods of third order for the numerical integration of the Schrödinger equation, Appl. Num. Anal. Comp. Math. 2(2)(2005) 238-244.
9. T.E. Simos, P-stable four-step exponentially-fitted method for the numerical integration of the Schrödinger equation, Comput. Lett. 1(1)(2005) 37-45
10. T.E. Simos, Eighth-order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem, J. Math. Chem. 21(1997) 359-372.
11. T.E. Simos, Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations, J. Math. Chem. 24(1998) 23-37.
12. T.E. Simos, A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 25(1999) 65-84.
13. G. Avdelas and T.E. Simos, Embedded eighth order methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 26(1999) 327-341.
14. T.E. Simos, A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation, J. Math. Chem. 27(2000) 343-356.
15. J. Vigo-Aguiar and T.E. Simos, A family of P-stable eighth algebraic order methods with exponential fitting facilities, J. Math. Chem. 29(2001) 177-189.
16. G. Avdelas, A. Konguetsof and T.E. Simos, A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 1. Development of the basic method, J. Math. Chem. 29(2001) 281-291.
17. G. Avdelas, A. Konguetsof and T.E. Simos, A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 2. Development of the generator, optimized generator and numerical results, J. Math. Chem. 29(2001) 293-305
18. T.E. Simos and J. Vigo-Aguiar, A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation, J. Math. Chem. 30(2001) 121-131.
19. T.E. Simos and J. Vigo-Aguiar, Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation, J. Math. Chem. 31(2002) 135-144
20. Z. Kalogiratou and T.E. Simos, Construction of trigonometrically and exponentially fotted Runge-Kutta-Nystrom methods for the numerical solution of the Schrödinger equation and related problems, J. Math. Chem. 31(2002) 211-232.
21. J. Vigo-Aguiar and T.E. Simos, Family of twelve steps exponentially fitting symmetric multistep methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 32(2002) 257-270.
22. G. Avdelas, E Kefalidis and T.E. Simos, New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation, J. Math. Chem. 31(2002) 371-404.
23. T.E. Simos, A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems J. Math. Chem. 34(2003) 39-58.
24. K. Tselios and T.E. Simos, Symplectic methods for the numerical solution of the radial shrodinger equation J. Math. Chem. 34(2003) 83-94
25. K. Tselios and T.E. Simos, Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation, J. Math. Chem. 35(2004) 55-63.
26. T.E. Simos, Exponentially fitted multiderivative methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 36(2004) 13-27.
27. D.P. Sakas and T.E. Simos, A family of multiderivative methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 37(2005) 317-331.
28. G. Psihoyios and T.E. Simos, Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrödinger equation, J. Math. Chem. 37(2005) 295-316.
29. Z. Anastassi and T.E. Simos, Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 37(2005) 281-293
30. Z. Kalogiratou, T. Monovasilis and T.E. Simos, Numerical solution of the two-dimensional time independent Schrödinger equation with Numerov-type methods, J. Math. Chem. 37(2005) 271-279.
31. T. Monovasilis, Z. Kalogiratou and T.E. Simos, Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation, J. Math. Chem. 37(2005) 263-270.
32. V. Arnold, Mathematical methods of Classical Mechanics, (Springer-Verlag, New York 1978).
33. X.S. Liu, X.Y. Liu, Z.Y. Zhou, P.Z. Ding and S.F. Pan, Numerical solution of the one-dimensional time-independent schrödinger equation by using symplectic schemes, International Journal of Quantum Chemistry 79(2000), 343-349.
34. J.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian Problem (Chapman and Hall, London, 1994).
35. H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150(1990) 262-268.
36. W. Zhu, X. Zhao and Y. Tang, Numerical methods with a high order of accuracy in the quantum system, J. Chem. Phys. 104(1996) 2275-2286.
37. A.D. Raptis and A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation, Comput. Phys. Commun., 14(1978) 1-5.
38. T.E. Simos and J.V. Aguiar, Exponentially fitted symplectic integrator, Phys. Rev. E, 67(2003) 16701.