41
Views
3
CrossRef citations to date
0
Altmetric
Original Articles

A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation

Pages 1095-1100 | Received 01 Oct 2005, Accepted 01 Oct 2005, Published online: 31 Jan 2007

References

  • Ixaru , L.Gr. and Micu , M. 1978 . Topics in Theoretical Physics , Bucharest : Central Institute of Physics .
  • Landau , L.D. and Lifshitz , F.M. 1965 . Quantum Mechanics , New York : Pergamon .
  • Prigogine , I. and Rice , Stuart , eds. 1997 . “ Advances in chemical physics. Vol.93: New methods in computational quantum mechanics ” . New York : John Wiley & Sons .
  • Herzberg , G. 1950 . Spectra of Diatomic Molecules , Toronto : Van Nostrand .
  • Simos , T.E. 2000 . “ Atomic structure computations ” . In Chemical Modelling: Applications and Theory , Edited by: Hinchliffe , A . 38 – 142 . Cambridge : The Royal Society of Chemistry .
  • Simos , T.E. 2002 . “ Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems ” . In Chemical Modelling: Application and Theory , Vol. 2 , 170 – 270 . Cambridge : The Royal Society of Chemistry .
  • Simos , T.E. and Williams , P.S. 1999 . On finite difference methods for the solution of the Schrödinger equation . Comput. Chem. , 23 : 513
  • T.E. Simos. Numerical solution of ordinary differential equations with periodical solution. Doctoral Dissertation, National Technical University of Athens, Greece (1990) (in Greek).
  • Berghe , G.V. , Van Daele , M. and Vande Vyver , H. 2004 . Exponentially-fitted algorithms: fixed or frequency dependent knot points? . Appl. Num. Anal. Comp. Math. , 1 ( 1 ) : 49
  • Monovasilis , Th. , Kalogiratou , Z. and Simos , T.E. 2004 . Numerical solution of the two-dimensional time independent Schrödinger equation by symplectic schemes . Appl. Num. Anal. Comp. Math. , 1 ( 1 ) : 195
  • Psihoyios , G. and Simos , T.E. 2004 . Effective numerical approximation of Schrödinger type equations through multiderivative exponentially-fitted schemes . Appl. Num. Anal. Comp. Math. , 1 ( 1 ) : 205
  • Psihoyios , G. and Simos , T.E. 2004 . Efficient numerical solution of orbital problems with the use of symmetric four-step trigonometrically-fitted methods . Appl. Num. Anal. Comp. Math. , 1 ( 1 ) : 216
  • Van Daele , M. and Vanden Berghe , G. 2004 . Extended one-step methods: An exponential fitting approach . Appl. Num. Anal. Comp. Math. , 1 ( 2 ) : 353
  • Vlachos , D.S. and Simos , T.E. 2004 . Partitioned linear multistep method for long term integration of the N-body problem . Appl. Num. Anal. Comp. Math. , 1 ( 2 ) : 540
  • Monovasilis , Th. and Kalogiratou , Z. 2005 . Trigonometrically and exponentially fitted symplectic methods of third order for the numerical integration of the Schrödinger equation . Appl. Num. Anal. Comp. Math. , 2 ( 2 ) : 238
  • Simos , T.E. 2005 . P-stable four-step exponentially-fitted method for the numerical integration of the Schrödinger equation . Comput. Lett. , 1 ( 1 ) : 37
  • Konguetsof , A. and Simos , T.E. 2001 . On the construction of exponentially-fitted methods for the numerical solution of the Schrödinger equation . J. Comput. Methods Sci. Eng. , 1 : 143
  • Simos , T.E. 1997 . Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem . J. Math. Chem. , 21 : 359
  • Simos , T.E. 1998 . Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrodinger equations . J. Math. Chem. , 24 : 23
  • Simos , T.E. 1999 . A family of P-stable exponentially-fitted methods for the numerical solution of the Schrodinger equation . J. Math. Chem. , 25 : 65
  • Avdelas , G. and Simos , T.E. 1999 . Embedded eighth order methods for the numerical solution of the Schrodinger equation . J. Math. Chem. , 26 : 327
  • Simos , T.E. 2000 . A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrodinger equation . J. Math. Chem. , 27 : 343
  • Vigo-Aguiar , Jesus and Simos , T.E. 2001 . A family of P-stable eighth algebraic order methods with exponential fitting facilities . J. Math. Chem. , 29 : 177
  • Avdelas , G. , Konguetsof , A. and Simos , T.E. 2001 . A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrodinger equation. Part 1. Development of the basic method . J. Math. Chem. , 29 : 281
  • Avdelas , G. , Konguetsof , A. and Simos , E. 2001 . A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrodinger equation. Part 2. Development of the generator, optimized generator and numerical results . J. Math. Chem. , 29 : 293
  • Simos , T.E. and Vigo-Aguiar , J. 2001 . A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrodinger equation . J. Math. Chem. , 30 : 121
  • Simos , T.E. and Vigo-Aguiar , J. 2002 . Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrodinger equation . J. Math. Chem. , 31 : 135
  • Kalogiratou , Z. and Simos , T.E. 2002 . Construction of trigonometrically and exponentially fotted Runge-Kutta-Nystrom methods for the numerical solution of the Schrodinger equation and related problems . J. Math. Chem. , 31 : 211
  • Vigo-Aguiar , J. and Simos , T.E. 2002 . Family of twelve steps exponentially fitting symmetric multistep methods for the numerical solution of the Schrodinger equation . J. Math. Chem. , 32 : 257
  • Avdelas , G. , Kefalidis , E. and Simos , T.E. 2002 . New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrodinger equation . J. Math. Chem. , 31 : 371
  • Simos , T.E. 2003 . A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrodinger equation and related problems . J. Math. Chem. , 34 : 39
  • Tselios , Kostas and Simos , T.E. 2003 . Symplectic methods for the numerical solution of the radial Shrodinger equation . J. Math. Chem. , 34 : 83
  • Raptis , A.D. and Allison , A.C. 1978 . Exponential—fitting methods for the numerical solution of the Schrödinger equation . Comput. Phys. Commun. , 14 : 1
  • Raptis , A.D. 1984 . Exponential multistep methods for ordinary differential equations . Bull. Greek Math. Soc. , 25 : 113
  • Ixaru , L.Gr. 1984 . Numerical Methods for Differential Equations and Applications , Dordrecht—Boston—Lancaster : Reidel .
  • Ixaru , L.Gr and Rizea , M. 1980 . A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies . Comput. Phys. Commun. , 19 : 23
  • Lambert , J.D. and Watson , I.A. 1976 . Symmetric multistep methods for periodic initial values problems . J. Inst. Math. Appl. , 18 : 189
  • Coleman , J.P. 1992 . “ Numerical methods for y″=f(x, y) ” . In Proceedings of the First International Colloquium on Numerical Analysis Edited by: Bainov , D. and Civachev , V. 27 – 38 . Bulgaria 1992
  • Coleman , J.P. 1989 . Numerical methods for y″=f(x, y) via rational approximation for the cosine . IMA J. Numer. Anal. , 9 : 145
  • Chawla , M.M. 1984 . Numerov made explicit has better stability . BIT , 24 : 117
  • Chawla , M.M. and Rao , P.S. 1986 . 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 : 329
  • Blatt , J.M. 1967 . Practical points concerning the solution of the Schrödinger equation . J. of Comput. Phys. , 1 : 382
  • Cooley , J.W. 1961 . An improved eigenvalue corrector formula for solving Schrödinger's equation for central fields . Math. Comp. , 15 : 363
  • Dormand , J.R. , El-Mikkawy , M.E. and Prince , P.J. 1987 . Families of Runge-Kutta-Nyström formulae . IMA Journal of Numerical Analysis , 7 : 423
  • Dormand , J.R. , El-Mikkawy , M.E.A. and Prince , P.J. 1987 . High-order embedded Runge-Kutta-Nyström formulae . IMA J. Numer. Anal. , 7 : 595
  • Simos , T.E. and Williams , P.S. 2002 . A new Runge-Kutta-Nystrom method with phase-lag of order infinity for the numerical solution of the Schrödinger equation . MATCH Commun. Math. Comput. Chem. , 45 : 123
  • Simos , T.E. 2004 . Multiderivative methods for the numerical solution of the Schrödinger equation . MATCH Commun. Math. Comput. Chem. , 45 : 7
  • Hairer , E. , Norsett , S.P. and Wanner , G. 1987 . Solving Ordinary Differential Equations I , Berlin : Springer-Verlag .
  • Allison , A.C. 1970 . The numerical solution of coupled differential equations arising from the Schrödinger equation . J. Comput. Phys. , 6 : 378
  • Berstein , R.B. , Dalgarno , A. , Massey , H. and Percival , C. 1963 . Thermal scattering of atoms by homonuclear diatomic molecules . Proc. R. Soc. Lond. Ser. A , 274 : 427
  • Berstrein , R.B. 1960 . Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams . J. Chem. Phys. , 33 : 795
  • Raptis , A.D. and Cash , J.R. 1985 . A variable step method for the numerical integration of the one-dimensional Schrödinger equation . Comput. Phys. Commun. , 36 : 113
  • Papakaliatakis , G. and Simos , T.E. 1999 . A new finite-difference method with minimal phase-lag for the numerical solution of differential equations with engineering applications . Adv. Eng. Software , 30 : 103
  • Raptis , A.D. and Simos , T. E. 1991 . A four-step phase-fitted method for the numerical integration of second order initial-value problem . BIT , 31 : 160
  • Raptis , A.D. 1983 . Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control . Comput. Phys. Commun. , 28 : 427
  • Raptis , A.D. 1981 . On the numerical solution of the Schrodinger equation . Comput. Phys. Commun. , 24 : 1
  • Kalogiratou , Zacharoula and Simos , T.E. 2000 . A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation . Appl. Math. Comput. , 112 : 99

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.