메뉴 건너뛰기




Volumn 31, Issue 4, 2002, Pages 371-404

New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation

Author keywords

Exponential fitting; Finite difference; Multistep methods; P stability; Scattering problems; Schr dinger equation

Indexed keywords

ARTICLE; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; MATHEMATICS;

EID: 0141608674     PISSN: 02599791     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021020705327     Document Type: Article
Times cited : (78)

References (64)
  • 2
    • 0002792006 scopus 로고
    • A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies
    • L.Gr. Ixaru and M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, Comput. Phys. Commun. 19 (1980) 23-27.
    • (1980) Comput. Phys. Commun. , vol.19 , pp. 23-27
    • Ixaru, L.Gr.1    Rizea, M.2
  • 3
    • 0004083934 scopus 로고
    • Pergamon Press, New York
    • L.D. Landau and F.M. Lifshitz, Quantum Mechanics (Pergamon Press, New York, 1965); R.L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, MA, 1980).
    • (1965) Quantum Mechanics
    • Landau, L.D.1    Lifshitz, F.M.2
  • 4
    • 0004085708 scopus 로고
    • Addison-Wesley, Reading, MA
    • L.D. Landau and F.M. Lifshitz, Quantum Mechanics (Pergamon Press, New York, 1965); R.L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, MA, 1980).
    • (1980) Introductory Quantum Mechanics
    • Liboff, R.L.1
  • 6
    • 0000981521 scopus 로고
    • Practical points concerning the solution of the Schrödinger equation
    • J.M. Blatt, Practical points concerning the solution of the Schrödinger equation, J. Comput. Phys. 1 (1967) 382-396.
    • (1967) J. Comput. Phys. , vol.1 , pp. 382-396
    • Blatt, J.M.1
  • 7
    • 0021505273 scopus 로고
    • A high order method for the numerical solution of the one-dimensional Schrödinger equation
    • J.R. Cash and A.D. Raptis, A high order method for the numerical solution of the one-dimensional Schrödinger equation, Comput. Phys. Commun. 33 (1984) 299-304.
    • (1984) Comput. Phys. Commun. , vol.33 , pp. 299-304
    • Cash, J.R.1    Raptis, A.D.2
  • 8
    • 0000335011 scopus 로고
    • A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation
    • J.R. Cash, A.D. Raptis and T.E. Simos, A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation, J. Comput. Phys. 91 (1990) 413-423.
    • (1990) J. Comput. Phys. , vol.91 , pp. 413-423
    • Cash, J.R.1    Raptis, A.D.2    Simos, T.E.3
  • 9
    • 84968515967 scopus 로고
    • An improved eigenvalue corrector formula for solving Schrödinger's equation for central fields
    • J.W. Cooley, An improved eigenvalue corrector formula for solving Schrödinger's equation for central fields, Math. Comp. 15 (1961) 363-374.
    • (1961) Math. Comp. , vol.15 , pp. 363-374
    • Cooley, J.W.1
  • 10
    • 36149029919 scopus 로고
    • Shooting methods for the Schrödinger equation
    • J. Killingbeck, Shooting methods for the Schrödinger equation, J. Phys. A 20 (1987) 1411-1417.
    • (1987) J. Phys. A , vol.20 , pp. 1411-1417
    • Killingbeck, J.1
  • 11
    • 84988121017 scopus 로고
    • On testing difference equations for the diatomic eigenvalue problem
    • H. Kobeissi and M. Kobeissi, On testing difference equations for the diatomic eigenvalue problem, J. Comput. Chem. 9 (1988) 844-850.
    • (1988) J. Comput. Chem. , vol.9 , pp. 844-850
    • Kobeissi, H.1    Kobeissi, M.2
  • 12
    • 0031357296 scopus 로고    scopus 로고
    • A family of hybrid exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation
    • R.M. Thomas and T.E. Simos, A family of hybrid exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation, J. Comput. Appl. Math. 87 (1997) 215-226.
    • (1997) J. Comput. Appl. Math. , vol.87 , pp. 215-226
    • Thomas, R.M.1    Simos, T.E.2
  • 13
    • 0034229908 scopus 로고    scopus 로고
    • Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation
    • G. Avdelas and T.E. Simos, Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation, Phys. Rev. E 62 (2000) 1375-1381.
    • (2000) Phys. Rev. E , vol.62 , pp. 1375-1381
    • Avdelas, G.1    Simos, T.E.2
  • 14
    • 0026868899 scopus 로고
    • The royal road to an energy-conserving predictor-corrector method
    • G.J. Kroes, The royal road to an energy-conserving predictor-corrector method, Comput. Phys. Commun. 70 (1992) 41-52.
    • (1992) Comput. Phys. Commun. , vol.70 , pp. 41-52
    • Kroes, G.J.1
  • 15
    • 0002481899 scopus 로고
    • On the numerical solution of the Schrödinger equation
    • A.D. Raptis, On the numerical solution of the Schrödinger equation, Comput. Phys. Commun. 24 (1981) 1-4.
    • (1981) Comput. Phys. Commun. , vol.24 , pp. 1-4
    • Raptis, A.D.1
  • 16
    • 0020458747 scopus 로고
    • Two-step methods for the numerical solution of the Schrödinger equation
    • A.D. Raptis, Two-step methods for the numerical solution of the Schrödinger equation, Computing 28 (1982) 373-378.
    • (1982) Computing , vol.28 , pp. 373-378
    • Raptis, A.D.1
  • 17
    • 0000172670 scopus 로고
    • Exponentially-fitted solutions of the eigenvalue Schrödinger equation with automatic error control
    • A.D. Raptis, Exponentially-fitted solutions of the eigenvalue Schrödinger equation with automatic error control, Comput. Phys. Commun. 28 (1983) 427-431.
    • (1983) Comput. Phys. Commun. , vol.28 , pp. 427-431
    • Raptis, A.D.1
  • 18
    • 0001176928 scopus 로고
    • Exponential multistep methods for ordinary differential equations
    • A.D. Raptis, Exponential multistep methods for ordinary differential equations, Bull. Greek Math. Soc. 25 (1984) 113-126.
    • (1984) Bull. Greek Math. Soc. , vol.25 , pp. 113-126
    • Raptis, A.D.1
  • 19
    • 0002481902 scopus 로고
    • Exponential-fitting methods for the numerical solution of the Schrödinger equation
    • 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.
    • (1978) Comput. Phys. Commun. , vol.14 , pp. 1-5
    • Raptis, A.D.1    Allison, A.C.2
  • 20
    • 0002779398 scopus 로고
    • Exponential and Bessel fitting methods for the numerical solution of the Schrödinger equation
    • A.D. Raptis and J.R. Cash, Exponential and Bessel fitting methods for the numerical solution of the Schrödinger equation, Comput. Phys. Commun. 44 (1987) 95-103.
    • (1987) Comput. Phys. Commun. , vol.44 , pp. 95-103
    • Raptis, A.D.1    Cash, J.R.2
  • 22
    • 0032842252 scopus 로고    scopus 로고
    • An exponentially-fitted eighth order method for the numerical solution of the Schrödinger equation
    • T.E. Simos, An exponentially-fitted eighth order method for the numerical solution of the Schrödinger equation, J. Comput. Appl. Math. 108 (1999) 177-194.
    • (1999) J. Comput. Appl. Math. , vol.108 , pp. 177-194
    • Simos, T.E.1
  • 23
    • 0033148222 scopus 로고    scopus 로고
    • Explicit eighth order methods for the numerical integration of initial-value problems with periodic or oscillating solutions
    • T.E. Simos, Explicit eighth order methods for the numerical integration of initial-value problems with periodic or oscillating solutions, Comput. Phys. Commun. 119 (1999) 32-44.
    • (1999) Comput. Phys. Commun. , vol.119 , pp. 32-44
    • Simos, T.E.1
  • 24
    • 0002895655 scopus 로고    scopus 로고
    • An eighth order exponentially-fitted method for the numerical integration of the Schrodinger equation
    • T.E. Simos, An eighth order exponentially-fitted method for the numerical integration of the Schrodinger equation, Comput. Chem. 22 (1998) 467-489.
    • (1998) Comput. Chem. , vol.22 , pp. 467-489
    • Simos, T.E.1
  • 25
    • 0000778465 scopus 로고
    • Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schrödinger equation
    • T.E. Simos, Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schrödinger equation, Phys. Lett. A 177 (1993) 345-350.
    • (1993) Phys. Lett. A , vol.177 , pp. 345-350
    • Simos, T.E.1
  • 26
    • 0000348458 scopus 로고
    • Numerical methods for solving radial Schrödinger equation
    • G. Vanden Berghe, V. Fack and H.E. De Meyer, Numerical methods for solving radial Schrödinger equation, J. Comput. Appl. Math. 29 (1989) 391-401.
    • (1989) J. Comput. Appl. Math. , vol.29 , pp. 391-401
    • Vanden Berghe, G.1    Fack, V.2    De Meyer, H.E.3
  • 27
    • 0001424633 scopus 로고    scopus 로고
    • A Numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation
    • T.E. Simos and G. Tougelidis, A Numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation, Comput. Chem. 20 (1996) 397.
    • (1996) Comput. Chem. , vol.20 , pp. 397
    • Simos, T.E.1    Tougelidis, G.2
  • 28
    • 0001176928 scopus 로고
    • Exponential multistep methods for ordinary differential equations
    • A.D. Raptis, Exponential multistep methods for ordinary differential equations, Bull. Greek Math. Soc. 25 (1984) 113-126.
    • (1984) Bull. Greek Math. Soc. , vol.25 , pp. 113-126
    • Raptis, A.D.1
  • 29
    • 0031219903 scopus 로고    scopus 로고
    • An explicit eighth order method with minimal phase-lag for the numerical solution of the Schrödinger equation
    • T.E. Simos and G. Tougelidis, An explicit eighth order method with minimal phase-lag for the numerical solution of the Schrödinger equation, Comput. Materials Sci. 8 (1997) 317-326.
    • (1997) Comput. Materials Sci. , vol.8 , pp. 317-326
    • Simos, T.E.1    Tougelidis, G.2
  • 30
    • 0004479705 scopus 로고    scopus 로고
    • Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems
    • T.E. Simos, Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems, Comput. Materials Sci. 18 (2000) 315-332.
    • (2000) Comput. Materials Sci. , vol.18 , pp. 315-332
    • Simos, T.E.1
  • 31
    • 0031326292 scopus 로고    scopus 로고
    • Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem
    • 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.
    • (1997) J. Math. Chem. , vol.21 , pp. 359-372
    • Simos, T.E.1
  • 32
    • 0039065106 scopus 로고    scopus 로고
    • Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations
    • 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.
    • (1998) J. Math. Chem. , vol.24 , pp. 23-37
    • Simos, T.E.1
  • 33
    • 0033429763 scopus 로고    scopus 로고
    • A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation
    • 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.
    • (1999) J. Math. Chem. , vol.25 , pp. 65-84
    • Simos, T.E.1
  • 34
    • 0033114512 scopus 로고    scopus 로고
    • Embedded eighth order methods for the numerical solution of the Schrödinger equation
    • 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.
    • (1999) J. Math. Chem. , vol.26 , pp. 327-341
    • Avdelas, G.1    Simos, T.E.2
  • 35
    • 0034338028 scopus 로고    scopus 로고
    • A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation
    • 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.
    • (2000) J. Math. Chem. , vol.27 , pp. 343-356
    • Simos, T.E.1
  • 36
    • 0141730856 scopus 로고    scopus 로고
    • Numerical methods for the solution of 1D, 2D and 3D differential equations arising in chemical problems
    • T.E. Simos, Numerical methods for the solution of 1D, 2D and 3D differential equations arising in chemical problems, Specialist Periodical Reports, Chemical Modelling: Applications and Theory 2 (2002) 170-268.
    • (2002) Specialist Periodical Reports, Chemical Modelling: Applications and Theory , vol.2 , pp. 170-268
    • Simos, T.E.1
  • 37
    • 0013350031 scopus 로고    scopus 로고
    • Atomic structure computations in chemical modelling
    • ed. A. Hinchliffe (The Royal Society of Chemistry)
    • T.E. Simos, Atomic structure computations in chemical modelling, in: Applications and Theory, UMIST, ed. A. Hinchliffe (The Royal Society of Chemistry, 2000) pp. 38-142.
    • (2000) Applications and Theory, UMIST , pp. 38-142
    • Simos, T.E.1
  • 38
    • 0035315407 scopus 로고    scopus 로고
    • A family of P-stable eighth algebraic order methods with exponential fitting facilities
    • 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.
    • (2001) J. Math. Chem. , vol.29 , pp. 177-189
    • Vigo-Aguiar, J.1    Simos, T.E.2
  • 39
    • 0035539374 scopus 로고    scopus 로고
    • 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
    • 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.
    • (2001) J. Math. Chem. , vol.29 , pp. 281-291
    • Avdelas, G.1    Konguetsof, A.2    Simos, T.E.3
  • 40
    • 0035539372 scopus 로고    scopus 로고
    • 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
    • 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 Schrodinger equation. Part 2. Development of the generator, optimized generator and numerical results, J. Math. Chem. 29 (2001) 293-305.
    • (2001) J. Math. Chem. , vol.29 , pp. 293-305
    • Avdelas, G.1    Konguetsof, A.2    Simos, T.E.3
  • 41
    • 0001237676 scopus 로고    scopus 로고
    • On finite difference methods for the solution of the Schrödinger equation
    • T.E. Simos and P.S. Williams, On finite difference methods for the solution of the Schrödinger equation, Comput. Chem. 23 (1999) 513-554.
    • (1999) Comput. Chem. , vol.23 , pp. 513-554
    • Simos, T.E.1    Williams, P.S.2
  • 42
    • 0005285043 scopus 로고    scopus 로고
    • P-stability and exponential-fitting methods for y″ = f (x, y)
    • J.P. Coleman and L.Gr. Ixaru, P-stability and exponential-fitting methods for y″ = f (x, y), IMA J. Numer. Anal. 16 (1996) 179-199.
    • (1996) IMA J. Numer. Anal. , vol.16 , pp. 179-199
    • Coleman, J.P.1    Ixaru, L.Gr.2
  • 43
    • 0034924872 scopus 로고    scopus 로고
    • New P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation
    • G. Avdelas, E. Kefalidis and T.E. Simos, New P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation, Comput. Materials Sci. 21 (2001) 301-319.
    • (2001) Comput. Materials Sci. , vol.21 , pp. 301-319
    • Avdelas, G.1    Kefalidis, E.2    Simos, T.E.3
  • 44
    • 34250466523 scopus 로고
    • Chebyshevian multistep methods for ordinary differential equations
    • T. Lyche, Chebyshevian multistep methods for ordinary differential equations, Numer. Math. 19 (1972) 65-75.
    • (1972) Numer. Math. , vol.19 , pp. 65-75
    • Lyche, T.1
  • 46
    • 0021316047 scopus 로고
    • Phase properties of high order, almost P-stable formulae
    • R.M. Thomas, Phase properties of high order, almost P-stable formulae, BIT 24 (1984) 225-238.
    • (1984) BIT , vol.24 , pp. 225-238
    • Thomas, R.M.1
  • 47
    • 77958409581 scopus 로고
    • Symmetric multistep methods for periodic initial value problems
    • J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976) 189-202.
    • (1976) J. Inst. Math. Appl. , vol.18 , pp. 189-202
    • Lambert, J.D.1    Watson, I.A.2
  • 48
    • 0016971912 scopus 로고
    • Solving nonstiff ordinary differential equations - The state of the art
    • L.F. Shampine, H.A. Watts and S.M. Davenport, Solving nonstiff ordinary differential equations - the state of the art, SIAM Rev. 18 (1976) 376-411.
    • (1976) SIAM Rev. , vol.18 , pp. 376-411
    • Shampine, L.F.1    Watts, H.A.2    Davenport, S.M.3
  • 49
    • 0034276333 scopus 로고    scopus 로고
    • A P-stable hybrid exponentially-fitted method for the numerical integration of the Schrödinger equiation
    • T.E. Simos and P.S. Williams, A P-stable hybrid exponentially-fitted method for the numerical integration of the Schrödinger equiation, Comput. Phys. Commun. 131 (2000) 109-119.
    • (2000) Comput. Phys. Commun. , vol.131 , pp. 109-119
    • Simos, T.E.1    Williams, P.S.2
  • 50
    • 0001752844 scopus 로고
    • A variable step method for the numerical integration of the one-dimensional Schrödinger equation
    • A.D. Raptis and J.R. Cash, A variable step method for the numerical integration of the one-dimensional Schrödinger equation, Comput. Phys. Commun. 36 (1985) 113-119.
    • (1985) Comput. Phys. Commun. , vol.36 , pp. 113-119
    • Raptis, A.D.1    Cash, J.R.2
  • 51
    • 0001401038 scopus 로고
    • The numerical solution of coupled differential equations arising from the Schrödinger equation
    • A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation, J. Comput. Phys. 6 (1970) 378-391.
    • (1970) J. Comput. Phys. , vol.6 , pp. 378-391
    • Allison, A.C.1
  • 53
    • 0000633647 scopus 로고
    • Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams
    • R.B. Berstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams, J. Chem. Phys. 33 (1960) 795-804.
    • (1960) J. Chem. Phys. , vol.33 , pp. 795-804
    • Berstein, R.B.1
  • 54
    • 0030367389 scopus 로고    scopus 로고
    • An eighth order method with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem
    • T.E. Simos, An eighth order method with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem, Internat. J. Modern Phys. C 7 (1996) 825-835.
    • (1996) Internat. J. Modern Phys. C , vol.7 , pp. 825-835
    • Simos, T.E.1
  • 55
    • 0000410643 scopus 로고    scopus 로고
    • Computer algebra programmes for the construction of a family of Numerov-type exponentially-fitted methods for the numerical solution of the Schrödinger equation
    • T.E. Simos and P.S. Williams, Computer algebra programmes for the construction of a family of Numerov-type exponentially-fitted methods for the numerical solution of the Schrödinger equation, Comput. Chem. 22 (1998) 185-218.
    • (1998) Comput. Chem. , vol.22 , pp. 185-218
    • Simos, T.E.1    Williams, P.S.2
  • 56
    • 0001590149 scopus 로고    scopus 로고
    • Bessel and Neuman-fitted methods for the numerical solution of the radial Schrödinger equation
    • T.E. Simos and P.S. Williams, Bessel and Neuman-fitted methods for the numerical solution of the radial Schrödinger equation, Comput. Chem. 21 (1997) 175-179.
    • (1997) Comput. Chem. , vol.21 , pp. 175-179
    • Simos, T.E.1    Williams, P.S.2
  • 57
    • 0002207932 scopus 로고    scopus 로고
    • Embedded methods for the numerical solution of the Schrödinger equation
    • G. Avdelas and T.E. Simos, Embedded methods for the numerical solution of the Schrödinger equation, Comput. Math. Appl. 31 (1996) 85-102.
    • (1996) Comput. Math. Appl. , vol.31 , pp. 85-102
    • Avdelas, G.1    Simos, T.E.2
  • 61
    • 0001330331 scopus 로고
    • Numerical methods for y″ = f (x, y) via rational approximation for the cosine
    • J.P. Coleman, Numerical methods for y″ = f (x, y) via rational approximation for the cosine, IMA J. Numer. Anal. 9 (1989) 145-165.
    • (1989) IMA J. Numer. Anal. , vol.9 , pp. 145-165
    • Coleman, J.P.1
  • 62
    • 0022752603 scopus 로고
    • A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II Explicit method
    • M.M. Chawla and P.S. Rao, 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) 329-337.
    • (1986) J. Comput. Appl. Math. , vol.15 , pp. 329-337
    • Chawla, M.M.1    Rao, P.S.2
  • 63
    • 0022789944 scopus 로고
    • Two-step fourth-order P-stable methods with phase-lag of order six for y″ = f (x, y)
    • M.M. Chawla, P.S. Rao and B. Neta, Two-step fourth-order P-stable methods with phase-lag of order six for y″ = f (x, y), J. Comput. Appl. Math. 16 (1986) 233-236.
    • (1986) J. Comput. Appl. Math. , vol.16 , pp. 233-236
    • Chawla, M.M.1    Rao, P.S.2    Neta, B.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.