Advertisement

Journal of Mathematical Chemistry

, Volume 50, Issue 7, pp 1861–1881 | Cite as

A new hybrid two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems

  • Ibraheem Alolyan
  • T. E. Simos
Original Paper

Abstract

The maximization of the efficiency of a hybrid two-step method for the numerical solution of the radial Schrödinger equation and related problems with periodic or oscillating solutions via the procedure of vanishing of the phase-lag and its derivatives is studied in this paper. More specifically, we investigate the vanishing of the phase-lag and its first and second derivatives and how this disappearance maximizes the efficiency of the hybrid two-step method.

Keywords

Numerical solution Schrödinger equation Multistep methods Hybrid methods Interval of periodicity P-stability Phase-lag Phase-fitted Derivatives of the phase-lag 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ixaru L.G., Micu M.: Topics in Theoretical Physics. Central Institute of Physics, Bucharest (1978)Google Scholar
  2. 2.
    Landau L.D., Lifshitz F.M.: Quantum Mechanics. Pergamon, New York (1965)Google Scholar
  3. 3.
    Prigogine, I., Rice, S. (eds): Advances in Chemical Physics Vol. 93: New Methods in Computational Quantum Mechanics. Wiley, New York (1997)Google Scholar
  4. 4.
    Herzberg G.: Spectra of Diatomic Molecules. Van Nostrand, Toronto (1950)Google Scholar
  5. 5.
    Simos T.E., Vigo-Aguiar J.: A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. J. Math. Chem. 30(1), 121–131 (2001)CrossRefGoogle Scholar
  6. 6.
    Tselios K., Simos T.E.: Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. J. Comput. Appl. Math. 175(1), 173–181 (2005)CrossRefGoogle Scholar
  7. 7.
    Anastassi Z.A., Simos T.E.: An optimized Runge-Kutta method for the solution of orbital problems. J. Comput. Appl. Math. 175(1), 1–9 (2005)CrossRefGoogle Scholar
  8. 8.
    Kosti A.A., Anastassi Z.A., Simos T.E.: An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 47(1), 315–330 (2010)CrossRefGoogle Scholar
  9. 9.
    Kalogiratou Z., Simos T.E.: Construction of trigonometrically and exponentially fitted Runge- Kutta-Nyström methods for the numerical solution of the Schrödinger equation and related problems a method of 8th algebraic order. J. Math. Chem. 31(2), 211–232 (2002)CrossRefGoogle Scholar
  10. 10.
    Simos T.E.: A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. IMA J. Numer. Anal. 21(4), 919–931 (2001)CrossRefGoogle Scholar
  11. 11.
    Simos T.E.: Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions. Appl. Math. Lett. 15(2), 217–225 (2002)CrossRefGoogle Scholar
  12. 12.
    Tsitouras C., Simos T.E.: Optimized Runge-Kutta pairs for problems with oscillating solutions. J. Comput. Appl. Math. 147(2), 397–409 (2002)CrossRefGoogle Scholar
  13. 13.
    Anastassi Z.A., Simos T.E.: Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 281–293 (2005)CrossRefGoogle Scholar
  14. 14.
    Anastassi Z.A., Simos T.E.: A family of exponentially-fitted Runge-Kutta methods with exponential order up to three for the numerical solution of the Schrödinger equation. J. Math. Chem. 41(1), 79–100 (2007)CrossRefGoogle Scholar
  15. 15.
    Lambert J.D., Watson I.A.: Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)CrossRefGoogle Scholar
  16. 16.
    Quinlan G.D., Tremaine S.: Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
  17. 17.
  18. 18.
    Avdelas G., Konguetsof A., Simos T.E.: 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(4), 281–291 (2001)CrossRefGoogle Scholar
  19. 19.
    Chawla M.M., Rao P.S.: An explicit sixth-order method with phase-lag of order eight for y′′ = f(t, y). J. Comput. Appl. Math. 17, 363–368 (1987)Google Scholar
  20. 20.
    Chawla M.M., Rao P.S.: An Noumerov-typ method with minimal phase-lag for the integration of second order periodic initial-value problems II explicit method. J. Comput. Appl. Math. 15, 329–337 (1986)CrossRefGoogle Scholar
  21. 21.
    Simos T.E., Williams P.S.: A finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189–205 (1997)CrossRefGoogle Scholar
  22. 22.
    Avdelas G., Konguetsof A., Simos T.E.: 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; optimization of the generator and numerical results. J. Math. Chem. 29(4), 293–305 (2001)CrossRefGoogle Scholar
  23. 23.
    Simos T.E., Vigo-Aguiar J.: Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation. J. Math. Chem. 31(2), 135–144 (2002)CrossRefGoogle Scholar
  24. 24.
    Konguetsof A., Simos T.E.: A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 93–106 (2003)CrossRefGoogle Scholar
  25. 25.
    Simos T.E., Famelis I.T., Tsitouras C.: Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27–40 (2003)CrossRefGoogle Scholar
  26. 26.
    Sakas D.P., Simos T.E.: Multiderivative methods of eighth algrebraic order with minimal phase- lag for the numerical solution of the radial Schrödinger equation. J. Comput. Appl. Math. 175(1), 161–172 (2005)CrossRefGoogle Scholar
  27. 27.
    Simos T.E.: Optimizing a class of linear multi-step methods for the approximate solution of the radial Schrödinger equation and related problems with respect to phase-lag. Cent. Eur. J. Phys. 9(6), 1518–1535 (2011)CrossRefGoogle Scholar
  28. 28.
    Sakas D.P., Simos T.E.: A family of multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 317–331 (2005)CrossRefGoogle Scholar
  29. 29.
    Van de Vyver H.: Phase-fitted and amplification-fitted two-step hybrid methods for y′′ = f(x, y). J. Comput. Appl. Math. 209(1), 33–53 (2007)CrossRefGoogle Scholar
  30. 30.
    Van de Vyver H.: An explicit Numerov-type method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53, 1339–1348 (2007)CrossRefGoogle Scholar
  31. 31.
    Simos T.E.: A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46(3), 981–1007 (2009)CrossRefGoogle Scholar
  32. 32.
    Alolyan I., Simos T.E.: High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation. J. Math. Chem. 48(4), 925–958 (2010)CrossRefGoogle Scholar
  33. 33.
    Alolyan I., Simos T.E.: Multistep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 48(4), 1092–1143 (2010)CrossRefGoogle Scholar
  34. 34.
    Alolyan I., Simos T.E.: A family of eight-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 49(3), 711–764 (2011)CrossRefGoogle Scholar
  35. 35.
    Stavroyiannis S., Simos T.E.: Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Numer. Math. 59(10), 2467–2474 (2009)CrossRefGoogle Scholar
  36. 36.
    T.E. Simos, Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag. J. Appl. Math. 2012. doi: 10.1155/2012/420387
  37. 37.
    Konguetsof A.: A new two-step hybrid method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871–890 (2010)CrossRefGoogle Scholar
  38. 38.
    Tselios K., Simos T.E.: Symplectic methods for the numerical solution of the radial Shrödinger equation. J. Math. Chem. 34(1–2), 83–94 (2003)CrossRefGoogle Scholar
  39. 39.
    Tselios K., Simos T.E.: Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation. J. Math. Chem. 35(1), 55–63 (2004)CrossRefGoogle Scholar
  40. 40.
    Monovasilis T., Simos T.E.: New second-order exponentially and trigonometrically fitted symplectic integrators for the numerical solution of the time-independent Schrödinger equation. J. Math. Chem. 42(3), 535–545 (2007)CrossRefGoogle Scholar
  41. 41.
    Monovasilis T., Kalogiratou Z., Simos T.E.: Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 37(3), 263–270 (2005)CrossRefGoogle Scholar
  42. 42.
    Monovasilis T., Kalogiratou Z., Simos T.E.: Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 40(3), 257–267 (2006)CrossRefGoogle Scholar
  43. 43.
    Kalogiratou Z., Monovasilis T., Simos T.E.: Symplectic integrators for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 83–92 (2003)CrossRefGoogle Scholar
  44. 44.
    Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae of high-order for long-time integration of orbital problems. Appl. Math. Lett. 22(10), 1616–1621 (2009)CrossRefGoogle Scholar
  45. 45.
    Kalogiratou Z., Simos T.E.: Newton-Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)CrossRefGoogle Scholar
  46. 46.
    Simos T.E.: High order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 209(1), 137–151 (2009)CrossRefGoogle Scholar
  47. 47.
    Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae for the solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 787–801 (2008)Google Scholar
  48. 48.
    Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae of high order for the numerical integration of the Schrödinger equation. J. Math. Chem. 44(2), 483–499 (2008)CrossRefGoogle Scholar
  49. 49.
    Simos T.E.: High-order closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Comput. Phys. Commun. 178(3), 199–207 (2008)CrossRefGoogle Scholar
  50. 50.
    Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae for numerical integration of the Schrödinger equation. Comput. Lett. 3(1), 45–57 (2007)CrossRefGoogle Scholar
  51. 51.
    Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. RevMexAA 42(2), 167–177 (2006)Google Scholar
  52. 52.
    Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae for long-time integration. Int. J. Mod. Phys. C 14(8), 1061–1074 (2003)CrossRefGoogle Scholar
  53. 53.
    Simos, T.E., New closed Newton-Cotes type formulae as multilayer symplectic integrators. J. Chem. Phys. 133(10), (104108-1–104108-7) (2010) (Article Number: 104108)Google Scholar
  54. 54.
    Vanden Berghe G., Van Daele M.: Exponentially fitted open Newton-Cotes differential methods as multilayer symplectic integrators. J. Chem. Phys. 132, 204107 (2010)CrossRefGoogle Scholar
  55. 55.
    Kalogiratou, Z., Monovasilis, T., T.E. Simos, A fifth-order symplectic trigonometrically fitted partitioned Runge-Kutta method. International Conference on Numerical Analysis and Applied Mathematics, Sept 16–20, 2007 Corfu, GREECE, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, vol. 936 (2007) pp. 313–317Google Scholar
  56. 56.
    Monovasilis T., Kalogiratou Z., Simos T.E.: Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems. Comput. Phys. Commun. 177(10), 757–763 (2007)CrossRefGoogle Scholar
  57. 57.
    Monovasilis T., Simos T.E.: Symplectic methods for the numerical integration of the Schrödinger equation. Comput. Mater. Sci. 38(3), 526–532 (2007)CrossRefGoogle Scholar
  58. 58.
    Monovasilis T., Kalogiratou Z., Simos T.E.: Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge-Kutta methods. Phys. Lett. A 372(5), 569–573 (2008)CrossRefGoogle Scholar
  59. 59.
    Monovasilis T., Kalogiratou Z., Simos T.E.: Symplectic partitioned Runge-Kutta methods with minimal phase-lag. Comput. Phys. Commun. 181(7), 1251–1254 (2010)CrossRefGoogle Scholar
  60. 60.
    Ixaru L.G., 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, 23–27 (1980)CrossRefGoogle Scholar
  61. 61.
    Raptis A.D., Allison A.C.: Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)CrossRefGoogle Scholar
  62. 62.
    Vigo-Aguiar J., Simos T.E.: Family of twelve steps exponential fitting symmetric multistep methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 32(3), 257–270 (2002)CrossRefGoogle Scholar
  63. 63.
    Psihoyios G., Simos T.E.: Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math. 158(1), 135–144 (2003)CrossRefGoogle Scholar
  64. 64.
    Psihoyios G., Simos T.E.: A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions. J. Comput. Appl. Math. 175(1), 137–147 (2005)CrossRefGoogle Scholar
  65. 65.
    Simos T.E.: Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004)CrossRefGoogle Scholar
  66. 66.
    Simos T.E.: Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Appl. Math. 110(3), 1331–1352 (2010)CrossRefGoogle Scholar
  67. 67.
    Avdelas G., Kefalidis E., Simos T.E.: New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 31(4), 371–404 (2002)CrossRefGoogle Scholar
  68. 68.
    Simos T.E.: A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems. J. Math. Chem. 34(1–2), 39–58 (2003)CrossRefGoogle Scholar
  69. 69.
    Simos T.E.: Exponentially—fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)CrossRefGoogle Scholar
  70. 70.
    Simos T.E.: A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem. 40(3), 305–318 (2006)CrossRefGoogle Scholar
  71. 71.
    Van de Vyver H.: A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems. Appl. Math. Comput. 189(1), 178–185 (2007)CrossRefGoogle Scholar
  72. 72.
    Simos T.E.: A family of four-step trigonometrically-fitted methods and its application to the Schrodinger equation. J. Math. Chem. 44(2), 447–466 (2009)CrossRefGoogle Scholar
  73. 73.
    Anastassi Z.A., Simos T.E.: A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution. J. Math. Chem. 45(4), 1102–1129 (2009)CrossRefGoogle Scholar
  74. 74.
    Psihoyios G., Simos T.E.: Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 37(3), 295–316 (2005)CrossRefGoogle Scholar
  75. 75.
    Psihoyios G., Simos T.E.: The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order predictor-corrector methods. J. Math. Chem. 40(3), 269–293 (2006)CrossRefGoogle Scholar
  76. 76.
    Wang Z.: P-stable linear symmetric multistep methods for periodic initial-value problems. Comput. Phys. Commun. 171(3), 162–174 (2005)CrossRefGoogle Scholar
  77. 77.
    Simos T.E.: A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation. J. Math. Chem. 27(4), 343–356 (2000)CrossRefGoogle Scholar
  78. 78.
    Anastassi Z.A., Simos T.E.: A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution. J. Math. Chem. 45(4), 1102–1129 (2009)CrossRefGoogle Scholar
  79. 79.
    Tang C., Wang W., Yan H., Chen Z.: High-order predictorcorrector of exponential fitting for the N-body problems. J. Comput. Phys. 214(2), 505–520 (2006)CrossRefGoogle Scholar
  80. 80.
    Panopoulos G.A., Anastassi Z.A., Simos T.E.: Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions. J. Math. Chem. 46(2), 604–620 (2009)CrossRefGoogle Scholar
  81. 81.
    Stavroyiannis S., Simos T.E.: Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Numer. Math. 59(10), 2467–2474 (2009)CrossRefGoogle Scholar
  82. 82.
    Stavroyiannis S., Simos T.E.: A nonlinear explicit two-step fourth algebraic order method of order infinity for linear periodic initial value problems. Comput. Phys. Commun. 181(8), 1362–1368 (2010)CrossRefGoogle Scholar
  83. 83.
    Anastassi Z.A., Simos T.E.: Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Rep. 482, 1–240 (2009)CrossRefGoogle Scholar
  84. 84.
    Vujasin R., Sencanski M., Radic-Peric J., Peric M.: A comparison of various variational approaches for solving the one-dimensional vibrational Schrödinger equation. MATCH Commun. Math. Comput. Chem. 63(2), 363–378 (2010)Google Scholar
  85. 85.
    Simos T.E., Williams P.S.: On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513–554 (1999)CrossRefGoogle Scholar
  86. 86.
    Ixaru L.G., Rizea M.: Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985)CrossRefGoogle Scholar
  87. 87.
    Vigo-Aguiar J., Simos T.E.: Review of multistep methods for the numerical solution of the radial Schrödinger equation. Int. J. Quantum Chem. 103(3), 278–290 (2005)CrossRefGoogle Scholar
  88. 88.
    Simos T.E., Zdetsis A.D., Psihoyios G., Anastassi Z.A.: Special issue on mathematical chemistry based on papers presented within ICCMSE 2005 preface. J. Math. Chem. 46(3), 727–728 (2009)CrossRefGoogle Scholar
  89. 89.
    Simos T.E., Psihoyios G., Anastassi Z.: Preface, proceedings of the international conference of computational methods in sciences and engineering 2005. Math. Comput. Model. 51(3-4), 137 (2010)CrossRefGoogle Scholar
  90. 90.
    Simos T.E., Psihoyios G.: Special issue: the international conference on computational methods in sciences and engineering 2004 - Preface. J. Comput. Appl. Math. 191(2), 165 (2006)CrossRefGoogle Scholar
  91. 91.
    T.E. Simos, Psihoyios, G., Special issue—selected papers of the international conference on computational methods in sciences and engineering (ICCMSE 2003) Kastoria, Greece, 12–16 September 2003—preface. J. Comput. Appl. Math. 175(1), IX (2005)Google Scholar
  92. 92.
    T.E. Simos, Vigo-Aguiar, J., Special issue—selected papers from the conference on computational and mathematical methods for science and engineering (CMMSE-2002)—Alicante University, Spain, 20–25 September 2002—preface. J. Comput. Appl. Math. 158(1), IX (2003)Google Scholar
  93. 93.
    T.E. Simos, Tsitouras C.: Special issue numerical methods in chemistry. MATCH Commun. Math. Comput. Chem. 60(3), 697–830 (2008)Google Scholar
  94. 94.
    T.E. Simos, Gutman, I., Papers presented on the international conference on computational methods in sciences and engineering (Castoria, Greece, September 12–16, 2003). MATCH Commun. Math. Comput. Chem 53(2), A3–A4 (2005)Google Scholar
  95. 95.
    Dormand J.R., El-Mikkawy M.E.A., Prince P.J.: Families of Runge-Kutta-Nyström formulae. IMA J. Numer. Anal. 7, 235–250 (1987)CrossRefGoogle Scholar
  96. 96.
    Dormand J.R., Prince P.J.: A family of embedded RungeKutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
  2. 2.Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and TechnologyUniversity of PeloponneseTripoliGreece
  3. 3.AthensGreece

Personalised recommendations