Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems

Computational Materials Science

Volumn 18, Issue 3-4, 2000, Pages 315-332

  Simos, T E ^a,b  

^a DEMOCRITUS UNIVERSITY OF THRACE   (Greece)

^b Amfithea Paleon Faliron   (Greece)

Author keywords

Coupled differential equations; Explicit methods; Exponential fitting; Hyperbolic equation; Initial value problems; Orbital problems; Oscillating solutions; Periodic solutions; Runge Kutta methods; Schr dinger equation

Indexed keywords

EID: 0004479705     PISSN: 09270256     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0927-0256(00)00112-9     Document Type: Article

References (20)

1. L.D. Landau, F.M. Lifshitz, Quantum Mechanics, Pergamon, New York, 1965.
2. I. Prigogine, Stuart Rice (Eds.), Advances in Chemical Physics, vol. 93: New Methods in Computational Quantum Mechanics, Wiley, New York, 1997.
3. J.R. Cash, A.D. Raptis, T.E. Simos, A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation, J. Comput. Phys. 91 (1990) 413.
4. T.E. Simos, A four-step method for the numerical solution of the Schrödinger equation, J. Comput. Appl. Math. 30 (1990) 251.
5. G. Avdelas, T.E. Simos, Embedded methods for the numerical solution of the Schrödinger equation, Comput. Math. Appl. 31 (1996) 85.
6. G. Avdelas, T.E. Simos, A generator of high-order embedded P-stable methods for the numerical solution of the Schrödinger equation, J. Comput. Appl. Math. 72 (1996) 345.
7. T.E. Simos, P-stable exponentially fitted methods for the numerical integration of the Schrödinger equation, J. Comput. Phys. 148 (1999) 305.
8. E. Hairer, S.P. Norset, G. Wanner, Solving Ordinary Differential Equations I (Nonstiff Problems), 2nd ed., Springer, Berlin, 1993.
9. T. Lyche, Chebyshevian multi-step methods for ordinary differential equations, Numer. Math. 19 (1972) 65.
10. A.D. Raptis, A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation, Comput. Phys. Commun. 14 (1978) 1.
11. J.W. Cooley, An improved eigenvalue corrector formula for solving Schrödinger's equation for central fields, Math. Comput. 15 (1961) 363.
12. J.M. Blatt, Practical points concerning the solution of the Schrödinger equation, J. Comput. Phys. 1 (1967) 382.
13. T.E. Simos, Numerical solution of ordinary differential equations with periodical solution, Doctoral Dissertation, National Technical University of Athens, 1990.
14. P.J. vanderHouwen, B.P. Sommeijer, Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal. 24 (1987) 595.
15. E. Stiefel, D.G. Bettis, Stabilization of Cowell's method, Numer. Math. 13 (1969) 154.
16. A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation, J. Comput. Phys. 6 (1970) 378-391.
17. R.B. Berstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules, Proc. Roy. Soc. London - Series A 274 (1963) 427-442.
18. R.B. Berstrein, Quantum mechanical phase shift analysis of differential elastic scattering of molecular beams, J. Chem. Phys. 33 (1960) 795-804.
19. A.D. Raptis, J.R. Cash, A variable step method for the numerical integration of the one-dimensional Schrödinger equation, Comput. Phys. Commun. 36 (1985) 113-119.
20. J.R. Dormand, P.J. Prince, A family of embedded Runge-Kutta formula, J. Comput. Appl. Math. 6 (1980) 19-26.