P-stable Exponentially Fitted Methods for the Numerical Integration of the Schrödinger Equation

Journal of Computational Physics

Volumn 148, Issue 2, 1999, Pages 305-321

  Simos, T E ^a  

^a DEMOCRITUS UNIVERSITY OF THRACE   (Greece)

Author keywords

Bound states problem; Exponentially fitted; Finite difference methods; Multistep methods; Phase shift; Radial Schr dinger equation; Resonance problem

Indexed keywords

BOUNDARY CONDITIONS; EIGENVALUES AND EIGENFUNCTIONS; NUMERICAL METHODS;

BOUND-STATES PROBLEMS; EXPONENTIALLY FITTED; EXPONENTIALLY-FITTED METHOD; FINITE-DIFFERENCE METHODS; MULTI STEP METHODS; NUMERICAL INTEGRATIONS; P-STABLE; PERIODIC SOLUTION; RADIAL SCHRÖDINGE EQUATION; RESONANCE PROBLEM;

FINITE DIFFERENCE METHOD;

EID: 0001160013     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1998.6103     Document Type: Article

References (37)

1. Blatt J. M. Practical points concerning the solution of the Schrödinger equation. J. Comput. Phys. 1:1967;382.
2. Cash J. R., Raptis A. D. A high order method for the numerical solution of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 33:1984;299.
3. Cash J. R., Raptis A. D., Simos T. E. A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation. J. Comput. Phys. 91:1990;413.
4. Coleman J. P. Numerical methods foryfxy. IMA J. Numer. Anal. 9:1989;145.
5. Cooley J. W. An improved eigenvalue corrector formula for solving Schrödinger's equation for central fields. Math. Comp. 15:1961;363.
6. Ixaru L. Gr., Micu M. Topics in Theoretical Physics. 1978.
7. Ixaru L. Gr., Rizea M. A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19:1980;23.
8. Killingbeck J. Shooting methods for the Schrödinger equation. J. Phys. A: Math. Gen. 20:1987;1411.
9. Kobeissi H., Kobeissi M. On testing difference equations for the diatomic eigenvalue problem. J. Comput. Chem. 9:1988;844.
10. Kobeissi H., Kobeissi M. A new variable step method for the numerical integration of the one-dimensional Schrödinger equation. J. Comput. Phys. 77:1988;501.
11. Kobeissi H., Kobeissi M., El-Hajj Ali. On computing eigenvalues of the Schrödinger equation for symmetrical potentials. J. Phys. A: Math. Gen. 22:1989;287.
12. Kroes G. J. The royal road to an energy-conserving predictor-corrector method. Comput. Phys. Commun. 70:1992;41.
13. Lambert J. D., Watson I. A. Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Appl. 18:1976;189.
14. Landau L. D., Lifshitz F. M. Quantum Mechanics. 1965.
15. Raptis A. D. On the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 24:1981;1.
16. Raptis A. D. Two-step methods for the numerical solution of the Schrödinger equation. Computing. 28:1982;373.
17. Raptis A. D. Exponentially-fitted solutions of the eigenvalue Schrödinger equation with automatic error control. Comput. Phys. Commun. 28:1983;427.
18. Raptis A. D. Exponential multistep methods for ordinary differential equations. Bull. Greek Math. Soc. 25:1984;113.
19. Raptis A. D., Allison A. C. Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14:1978;1.
20. Raptis A. D., Cash J. R. Exponential and Bessel fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 44:1987;95.
21. Simos T. E. Numerical Solution of Ordinary Differential Equations with Periodical Solution. 1990.
22. Simos T. E. A four-step method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 30:1990;251.
23. Simos T. E. Some new four-step exponential-fitting methods for the numerical solution of the radial Schrödinger equation. IMA J. Numer. Anal. 11:1991;347.
24. Simos T. E. Exponential fitted methods for the numerical integration of the Schrödinger equation. Comput. Phys. Commun. 71:1992;32.
25. Simos T. E. Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schrödinger equation. Phys. Lett. A. 177:1993;345.
26. Vanden Berghe G., Fack V., De Meyer H. E. Numerical methods for solving radial Schrödinger equation. J. Comput. Appl. Math. 29:1989;391.
27. Herzberg G. Spectra of Diatomic Molecules. 1950.
28. Simos T. E., Tougelidis G. A Numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation. Comput. & Chem. 20:1996;397.
29. Henrici P. Discrete Variable Methods in Ordinary Differential Equations. 1962.
30. Lyche T. Chebyshevian multistep methods for ordinary differential equations. Numer. Math. 19:1972;65.
31. Raptis A. D. Exponential multistep methods for ordinary differential equations. Bull. Greek Math. Soc. 25:1984;113.
32. Chawla M. M., Rao P. S. A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II. Explicit method. J. Comput. Appl. Math. 15:1986;329.
33. Chawla M. M., Rao P. S., Neta B. Two-step fourth-order P-stable methods with phase-lag of order six foryfxy. J. Comput. Appl. Math. 16:1986;233.
34. Coleman J. P., Ixaru L. Gr. P-stability and exponential-fitting methods foryfxy. IMA J. Numer. Anal. 16:1996;179.
35. Thomas R. M. Phase properties of high order, almost P-stable formulae. BIT. 24:1984;225.
36. Dormand J. R., Prince P. J. Runge-Kutta-Nyström triples. Comput. Math. Appl. 13:1987;937.
37. Dormand J. R., El-Mikkawy M. E., Prince P. J. Families of Runge-Kutta-Nyström formulae. IMA J. Numer. Anal. 7:1987;423.