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Journal of Mathematical Chemistry

, Volume 55, Issue 8, pp 1521–1547 | Cite as

An efficient six-step method for the solution of the Schrödinger equation

  • Dmitriy B. Berg
  • T. E. SimosEmail author
Original Paper

Abstract

In this paper we develop an efficient six-step method for the solution of the Schrödinger equation and related problems. The characteristics of the new obtained scheme are:
  • It is of twelfth algebraic order.

  • It has three stages.

  • It has vanished phase-lag.

  • It has vanished its derivatives up to order two.

  • All the stages of the scheme are approximations on the point \(x_{n+3}\).

This method is developed for the first time in the literature. A detailed theoretical analysis of the method is also presented. In the theoretical analysis, a comparison with the the classical scheme of the family (i.e. scheme with constant coefficients) and with recently developed algorithm of the family with eliminated phase-lag and its first derivative is also given. Finally, we study the accuracy and computational effectiveness of the new developed algorithm for the on the approximation of the solution of the Schrödinger equation. The above analysis which is described in this paper, leads to the conclusion that the new algorithm is more efficient than other known or recently obtained schemes of the literature.

Keywords

Schrödinger equation Multistep methods Multistage methods Interval of periodicity Phase-lag Phase-fitted Derivatives of the phase-lag 

Mathematics Subject Classification

65L05 

Notes

Author contributions

Authors whose names appear on the submission have contributed sufficiently to the scientific work and therefore share collective responsibility and accountability for the results.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Informed consent

Consent to submit has been received explicitly from all co-authors, as well as from the responsible authorities—tacitly or explicitly—at the institute/organization where the work has been carried out, before the work is submitted.

Supplementary material

10910_2017_742_MOESM1_ESM.pdf (192 kb)
Supplementary material 1 (pdf 192 KB)

References

  1. 1.
    T.E. Simos, J. Vigo-Aguiar, 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
  2. 2.
    K. Tselios, T.E. Simos, 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
  3. 3.
    Z.A. Anastassi, T.E. Simos, An optimized Runge–Kutta method for the solution of orbital problems. J. Comput. Appl. Math. 175(1), 1–9 (2005)CrossRefGoogle Scholar
  4. 4.
    D.F. Papadopoulos, T.E. Simos, A new methodology for the construction of optimized Runge–Kutta–Nyström Methods. Int. J. Mod. Phys. C 22(6), 623–634 (2011)CrossRefGoogle Scholar
  5. 5.
    D.F. Papadopoulos, T.E. Simos, A modified Runge–Kutta–Nyström method by using phase lag properties for the numerical solution of orbital problems. Appl. Math. Inf. Sci. 7(2), 433–437 (2013)CrossRefGoogle Scholar
  6. 6.
    D.F. Papadopoulos, T.E. Simos, The use of phase lag and amplification error derivatives for the construction of a modified Runge–Kutta–Nyström method. Abstr. Appl. Anal. 2013, 910624 (2013). doi: 10.1155/2013/910624 CrossRefGoogle Scholar
  7. 7.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7(1), 81–85 (2013)CrossRefGoogle Scholar
  8. 8.
    A.A. Kosti, Z.A. Anastassi, T.E. Simos, Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems. Comput. Math. Appl. 61(11), 3381–3390 (2011)CrossRefGoogle Scholar
  9. 9.
    Z. Kalogiratou, T. Monovasilis, G. Psihoyios, T.E. Simos, Runge–Kutta type methods with special properties for the numerical integration of ordinary differential equations. Phys. Rep. Rev. Sect. Phys. Lett. 536(3), 75–146 (2014)Google Scholar
  10. 10.
    Z. Kalogiratou, T. Monovasilis, T.E. Simos, A fourth order modified trigonometrically fitted symplectic Runge–Kutta–Nyström method. Comput. Phys. Commun. 185(12), 3151–3155 (2014)CrossRefGoogle Scholar
  11. 11.
    A.A. Kosti, Z.A. Anastassi, T.E. Simos, 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
  12. 12.
    Z. Kalogiratou, T.E. Simos, 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
  13. 13.
    T.E. Simos, 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
  14. 14.
    T.E. Simos, 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
  15. 15.
    C. Tsitouras, T.E. Simos, Optimized Runge–Kutta pairs for problems with oscillating solutions. J. Comput. Appl. Math. 147(2), 397–409 (2002)CrossRefGoogle Scholar
  16. 16.
    Z.A. Anastassi, T.E. Simos, Trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 281–293 (2005)CrossRefGoogle Scholar
  17. 17.
    Z.A. Anastassi, T.E. Simos, 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
  18. 18.
    J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)CrossRefGoogle Scholar
  19. 19.
    G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
  20. 20.
    H. Ramos, Z. Kalogiratou, T. Monovasilis, T.E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms 72, 1089–1102 (2016)CrossRefGoogle Scholar
  21. 21.
    C. Tsitouras, I.T. Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)CrossRefGoogle Scholar
  22. 22.
    Ch. Tsitouras, I.T. Famelis, T.E. Simos, Phase-fitted Runge–Kutta pairs of orders 8(7). J. Comput. Appl. Math. 321, 226–231 (2017)Google Scholar
  23. 23.
  24. 24.
    G. Avdelas, A. Konguetsof, 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(4), 281–291 (2001)CrossRefGoogle Scholar
  25. 25.
    M.M. Chawla, P.S. Rao, An explicit sixth-order method with phase-lag of order eight for \(y^{\prime \prime }=f(t, y)\). J. Comput. Appl. Math. 17, 363–368 (1987)Google Scholar
  26. 26.
    M.M. Chawla, P.S. Rao, An Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems II Explicit Method. J. Comput. Appl. Math. 15, 329–337 (1986)CrossRefGoogle Scholar
  27. 27.
    T.E. Simos, P.S. Williams, A finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189–205 (1997)CrossRefGoogle Scholar
  28. 28.
    G. Avdelas, A. Konguetsof, T.E. Simos, 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
  29. 29.
    T.E. Simos, J. Vigo-Aguiar, 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
  30. 30.
    A. Konguetsof, T.E. Simos, 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
  31. 31.
    T.E. Simos, I.T. Famelis, C. Tsitouras, Zero dissipative. Explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27–40 (2003)CrossRefGoogle Scholar
  32. 32.
    D.P. Sakas, T.E. Simos, 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
  33. 33.
    T.E. Simos, 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. Central Eur. J. Phys. 9(6), 1518–1535 (2011)Google Scholar
  34. 34.
    D.P. Sakas, T.E. Simos, A family of multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 317–331 (2005)CrossRefGoogle Scholar
  35. 35.
    H. Van de Vyver, Phase-fitted and amplification-fitted two-step hybrid methods for \(y^{\prime \prime }=f(x, y)\). J. Comput. Appl. Math. 209(1), 33–53 (2007)CrossRefGoogle Scholar
  36. 36.
    H. Van de Vyver, An explicit Numerov-type method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53, 1339–1348 (2007)CrossRefGoogle Scholar
  37. 37.
    T.E. Simos, A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46(3), 981–1007 (2009)CrossRefGoogle Scholar
  38. 38.
    G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A new symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems. Int. J. Mod. Phys. C 22(2), 133–153 (2011)CrossRefGoogle Scholar
  39. 39.
    G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions. Comput. Phys. Commun. 182(8), 1626–1637 (2011)CrossRefGoogle Scholar
  40. 40.
    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. Article ID 420387, Volume 2012 (2012)Google Scholar
  41. 41.
    T.E. Simos, A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 49(10), 2486–2518 (2011)CrossRefGoogle Scholar
  42. 42.
    I. Alolyan, T.E. Simos, A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 62(10), 3756–3774 (2011)CrossRefGoogle Scholar
  43. 43.
    I. Alolyan, T.E. Simos, A new four-step hybrid type method with vanished phase-lag and its first derivatives for each level for the approximate integration of the Schrödinger equation. J. Math. Chem. 51, 2542–2571 (2013)CrossRefGoogle Scholar
  44. 44.
    I. Alolyan, T.E. Simos, A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem. 52, 917–947 (2014)CrossRefGoogle Scholar
  45. 45.
    I. Alolyan, T.E. Simos, Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53, 1808–1834 (2015)CrossRefGoogle Scholar
  46. 46.
    I. Alolyan, T.E. Simos, Family of symmetric linear six-step methods with vanished phase-lag and its derivatives and their application to the radial Schrödinger equation and related problems. J. Math. Chem. 54, 466–502 (2016)CrossRefGoogle Scholar
  47. 47.
    I. Alolyan, T.E. Simos, A family of two stages tenth algebraic order symmetric six-step methods with vanished phase-lag and its first derivatives for the numerical solution of the Radial Schrödinger equation and related problems. J. Math. Chem. 54(9), 1835–1862 (2016)CrossRefGoogle Scholar
  48. 48.
    I. Alolyan, T.E. Simos, A new two stages tenth algebraic order symmetric six-step method with vanished phase-lag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems. J. Math. Chem. 55(1), 105–131 (2017)CrossRefGoogle Scholar
  49. 49.
    I. Alolyan, T.E. Simos, New two stages high order symmetric six-step method with vanished phase-lag and its first, second and third derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 55(2), 503–531 (2017)CrossRefGoogle Scholar
  50. 50.
    I. Alolyan, Z.A. Anastassi, T.E. Simos, A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput. 218(9), 5370–5382 (2012)Google Scholar
  51. 51.
    Z.A. Anastassi, T.E. Simos, A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems. J. Comput. Appl. Math. 236(16), 3880–3889 (2012)CrossRefGoogle Scholar
  52. 52.
    G.A. Panopoulos, T.E. Simos, An optimized symmetric 8-step semi-embedded predictor-corrector method for IVPs with oscillating solutions. Appl. Math. Inf. Sci. 7(1), 73–80 (2013)CrossRefGoogle Scholar
  53. 53.
    G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A new eight-step symmetric embedded predictor-corrector method (EPCM) for orbital problems and related IVPs with oscillatory solutions. Astron. J. 145(3), (2013). Article Number: 75 doi: 10.1088/0004-6256/145/3/75
  54. 54.
    T.E. Simos, New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. Part I: construction and theoretical analysis. J. Math. Chem. 51(1), 194–226 (2013)CrossRefGoogle Scholar
  55. 55.
    T.E. Simos, On the explicit four-step methods with vanished phase-lag and its first derivative. Appl. Math. Inf. Sci. 8(2), 447–458 (2014)CrossRefGoogle Scholar
  56. 56.
    G.A. Panopoulos, T.E. Simos, A new optimized symmetric embedded predictor–corrector method (EPCM) for initial-value problems with oscillatory solutions. Appl. Math. Inf. Sci. 8(2), 703–713 (2014)CrossRefGoogle Scholar
  57. 57.
    T.E. Simos, An explicit four-step method with vanished phase-lag and its first and second derivatives. J. Math. Chem. 52(3), 833–855 (2014)CrossRefGoogle Scholar
  58. 58.
    T.E. Simos, An explicit linear six-step method with vanished phase-lag and its first derivative. J. Math. Chem. 52(7), 1895–1920 (2014)CrossRefGoogle Scholar
  59. 59.
    T.E. Simos, A new explicit hybrid four-step method with vanished phase-lag and its derivatives. J. Math. Chem. 52(7), 1690–1716 (2014)CrossRefGoogle Scholar
  60. 60.
    I. Alolyan, T.E. Simos, A family of explicit linear six-step methods with vanished phase-lag and its first derivative. J. Math. Chem. 52(8), 2087–2118 (2014)CrossRefGoogle Scholar
  61. 61.
    I. Alolyan, T.E. Simos, A hybrid type four-step method with vanished phase-lag and its first, second and third derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem. 52(9), 2334–2379 (2014)CrossRefGoogle Scholar
  62. 62.
    T.E. Simos, Multistage symmetric two-step P-stable method with vanished phase-lag and its first, second and third derivatives. Appl. Comput. Math. 14(3), 296–315 (2015)Google Scholar
  63. 63.
    F. Hui, T.E. Simos, Four stages symmetric two-step P-stable method with vanished phase-lag and its first, second, third and fourth derivatives. Appl. Comput. Math. 15(2), 220–238 (2016)Google Scholar
  64. 64.
    I. Alolyan, T.E. Simos, A family of embedded explicit six-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation: development and theoretical analysis. J. Math. Chem. 54(5), 1159–1186 (2016)CrossRefGoogle Scholar
  65. 65.
    M. Liang, T.E. Simos, A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation. J. Math. Chem. 54(5), 1187–1211 (2016)CrossRefGoogle Scholar
  66. 66.
    I. Alolyan, T.E. Simos, An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems. J. Math. Chem. 54(4), 1010–1040 (2016)CrossRefGoogle Scholar
  67. 67.
    Z. Zhou, T.E. Simos, A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 54(2), 442–465 (2016)CrossRefGoogle Scholar
  68. 68.
    F. Hui, T.E. Simos, A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(10), 2191–2213 (2015)CrossRefGoogle Scholar
  69. 69.
    I. Alolyan, T.E. Simos, A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 53(8), 1915–1942 (2015)CrossRefGoogle Scholar
  70. 70.
    I. Alolyan, T.E. Simos, A high algebraic order predictor–corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 53(7), 1495–1522 (2015)CrossRefGoogle Scholar
  71. 71.
    K. Mu, T.E. Simos, A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation. J. Math. Chem. 53(5), 1239–1256 (2015)CrossRefGoogle Scholar
  72. 72.
    I. Alolyan, T.E. Simos, A predictor-corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(2), 685–717 (2015)CrossRefGoogle Scholar
  73. 73.
    I. Alolyan, T.E. Simos, A new eight algebraic order embedded explicit six-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 54(8), 1696–1727 (2016)CrossRefGoogle Scholar
  74. 74.
    T.E. Simos, A new explicit four-step method with vanished phase-lag and its first and second derivatives. J. Math. Chem. 53(1), 402–429 (2015)CrossRefGoogle Scholar
  75. 75.
    G.A. Panopoulos, T.E. Simos, An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math. 290, 1–15 (2015)CrossRefGoogle Scholar
  76. 76.
    A. Konguetsof, A new two-step hybrid method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871–890 (2010)CrossRefGoogle Scholar
  77. 77.
    K. Tselios, T.E. Simos, Symplectic methods for the numerical solution of the radial Shrödinger equation. J. Math. Chem. 34(1–2), 83–94 (2003)CrossRefGoogle Scholar
  78. 78.
    K. Tselios, T.E. Simos, Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation. J. Math. Chem. 35(1), 55–63 (2004)CrossRefGoogle Scholar
  79. 79.
    T. Monovasilis, T.E. Simos, 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
  80. 80.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 37(3), 263–270 (2005)CrossRefGoogle Scholar
  81. 81.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, 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
  82. 82.
    Z. Kalogiratou, T. Monovasilis, T.E. Simos, Symplectic integrators for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 83–92 (2003)CrossRefGoogle Scholar
  83. 83.
    T.E. Simos, 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
  84. 84.
    Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)CrossRefGoogle Scholar
  85. 85.
    T.E. Simos, High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 209(1), 137–151 (2009)Google Scholar
  86. 86.
    T.E. Simos, 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
  87. 87.
    T.E. Simos, 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
  88. 88.
    T.E. Simos, 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
  89. 89.
    T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae for numerical integration of the Schrödinger equation. Comput. Lett. 3(1), 45–57 (2007)CrossRefGoogle Scholar
  90. 90.
    T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. RevMexAA 42(2), 167–177 (2006)Google Scholar
  91. 91.
    T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae for long-time integration. Int. J. Mod. Phys. C 14(8), 061–1074 (2003)CrossRefGoogle Scholar
  92. 92.
    T.E. Simos, New closed Newton–Cotes type formulae as multilayer symplectic integrators. J. Chem. Phys. 133(10), Article Number: 104108 (2010)Google Scholar
  93. 93.
    T.E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration. Abstr. Appl. Anal. Article Number: 182536, (2012). doi: 10.1155/2012/182536
  94. 94.
    T.E. Simos, High order closed Newton–Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation. J. Math. Chem. 50(5), 1224–1261 (2012)CrossRefGoogle Scholar
  95. 95.
    T.E. Simos, accurately closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Int. J. Mod. Phys. C 24(3), 1350014 (2013). doi: 10.1142/S0129183113500149
  96. 96.
    T.E. Simos, New open modified Newton Cotes type formulae as multilayer symplectic integrators. Appl. Math. Model. 37(4), 1983–1991 (2013)CrossRefGoogle Scholar
  97. 97.
    G. Vanden Berghe, M. Van Daele, Exponentially fitted open Newton–Cotes differential methods as multilayer symplectic integrators. J. Chem. Phys. 132, 204107 (2010)Google Scholar
  98. 98.
    Z. Kalogiratou, T. Monovasilis, T.E. Simos, A fifth-order symplectic trigonometrically fitted partitioned Runge–Kutta method, International Conference on Numerical Analysis and Applied Mathematics, SEP 16-20, 2007 Corfu, GREECE, Numerical Analysis and Applied Mathematics. AIP Conf. Proc. 936, 313–317 (2007)Google Scholar
  99. 99.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, 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
  100. 100.
    T. Monovasilis, T.E. Simos, Symplectic methods for the numerical integration of the Schrödinger equation. Comput. Mater. Sci. 38(3), 526–532 (2007)CrossRefGoogle Scholar
  101. 101.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, 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
  102. 102.
    Z. Kalogiratou, T. Monovasilis, T.E. Simos, New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation. Comput. Math. Appl. 60(6), 1639–1647 (2010)Google Scholar
  103. 103.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, A family of trigonometrically fitted partitioned Runge–Kutta symplectic methods. Appl. Math. Comput. 209(1), 91–96 (2009)Google Scholar
  104. 104.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Construction of exponentially fitted symplectic Runge–Kutta–Nyström methods from partitioned Runge–Kutta methods. Mediterr. J. Math. 13(4), 2271–2285 (2016)CrossRefGoogle Scholar
  105. 105.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Two new phase-fitted symplectic partitioned Runge–Kutta methods. Int. J. Mod. Phys. C 22(12), 1343–1355 (2011)CrossRefGoogle Scholar
  106. 106.
    K. Tselios, T.E. Simos, Optimized fifth order symplectic integrators for orbital problems. Revista Mexicana de Astronomia y Astrofisica 49(1), 11–24 (2013)Google Scholar
  107. 107.
    Z. Kalogiratou, T. Monovasilis, H. Ramos, T.E. Simos, A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math. 303, 146–155 (2016)CrossRefGoogle Scholar
  108. 108.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Symplectic partitioned Runge–Kutta methods with minimal phase-lag. Comput. Phys. Commun. 181(7), 1251–1254 (2010)CrossRefGoogle Scholar
  109. 109.
    L.G. Ixaru, 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, 23–27 (1980)CrossRefGoogle Scholar
  110. 110.
    A.D. Raptis, A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)CrossRefGoogle Scholar
  111. 111.
    J. Vigo-Aguiar, T.E. Simos, 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
  112. 112.
    G. Psihoyios, T.E. Simos, Trigonometrically fitted predictor–corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math. 158(1), 135–144 (2003)CrossRefGoogle Scholar
  113. 113.
    G. Psihoyios, T.E. Simos, 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
  114. 114.
    T.E. Simos, Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004)CrossRefGoogle Scholar
  115. 115.
    T.E. Simos, Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Appl. Math. 110(3), 1331–1352 (2010)CrossRefGoogle Scholar
  116. 116.
    G. Avdelas, E. Kefalidis, T.E. Simos, 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
  117. 117.
    T.E. Simos, 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
  118. 118.
    T.E. Simos, Exponentially-fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)CrossRefGoogle Scholar
  119. 119.
    T.E. Simos, A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem. 40(3), 305–318 (2006)CrossRefGoogle Scholar
  120. 120.
    H. Van de Vyver, A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems. Appl. Math. Comput. 189(1), 178–185 (2007)Google Scholar
  121. 121.
    T.E. Simos, 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
  122. 122.
    Z.A. Anastassi, T.E. Simos, 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
  123. 123.
    G. Psihoyios, T.E. Simos, 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
  124. 124.
    G. Psihoyios, T.E. Simos, 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
  125. 125.
    Z. Wang, P-stable linear symmetric multistep methods for periodic initial-value problems. Comput. Phys. Commun. 171(3), 162–174 (2005)CrossRefGoogle Scholar
  126. 126.
    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(4), 343–356 (2000)CrossRefGoogle Scholar
  127. 127.
    C. Tang, W. Wang, H. Yan, Z. Chen, High-order predictor–corrector of exponential fitting for the N-body problems. J. Comput. Phys. 214(2), 505–520 (2006)CrossRefGoogle Scholar
  128. 128.
    G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions. J. Math. Chem. 46(2), 604–620 (2009)CrossRefGoogle Scholar
  129. 129.
    S. Stavroyiannis, T.E. Simos, 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
  130. 130.
    S. Stavroyiannis, T.E. Simos, 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
  131. 131.
    Z.A. Anastassi, T.E. Simos, Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Rep. 482, 1–240 (2009)CrossRefGoogle Scholar
  132. 132.
    R. Vujasin, M. Sencanski, J. Radic-Peric, M. Peric, 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
  133. 133.
    T.E. Simos, P.S. Williams, On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513–554 (1999)CrossRefGoogle Scholar
  134. 134.
    L.G. Ixaru, M. Rizea, Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985)CrossRefGoogle Scholar
  135. 135.
    J. Vigo-Aguiar, T.E. Simos, Review of multistep methods for the numerical solution of the radial Schrödinger equation. Int. J. Quantum Chem. 103(3), 278–290 (2005)CrossRefGoogle Scholar
  136. 136.
    M.A. Medvedev, T.E. Simos, Two stages six-step method with eliminated phase-lag and its first, second, third and fourth derivatives for the approximation of the Schrödinger equation. J. Math. Chem. 55(4), 961–986 (2017)CrossRefGoogle Scholar
  137. 137.
    D.B. Berg, T.E. Simos, High order computationally economical six-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 55(4), 987–1013 (2017)CrossRefGoogle Scholar
  138. 138.
    D.B. Berg, T.E. Simos, Three stages symmetric six-step method with eliminated phase-lag and its derivatives for the solution of the Schrödinger equation. J. Math. Chem. (2017). doi: 10.1007/s10910-017-0738-8 Google Scholar
  139. 139.
    S. Kottwitz, LaTeX Cookbook, pp. 231–236 (Packt Publishing Ltd., Birmingham B3 2PB, UK 2015)Google Scholar
  140. 140.
    L.G. Ixaru, M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)Google Scholar
  141. 141.
    J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of Runge–Kutta–Nyström formulae. IMA J. Numer. Anal. 7, 235–250 (1987)CrossRefGoogle Scholar
  142. 142.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
  143. 143.
    L.D. Landau, F.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965)Google Scholar
  144. 144.
    I. Prigogine (Eds): Advances in Chemical Physics Vol. 93: New Methods in Stuart Rice Computational Quantum Mechanics (Wiley, 1997)Google Scholar
  145. 145.
    G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, Toronto, 1950)Google Scholar
  146. 146.
    T.E. Simos, G. Psihoyios, Special issue: the international conference on computational methods in sciences and engineering 2004—preface. J. Comput. Appl. Math. 191(2), 165–165 (2006)CrossRefGoogle Scholar
  147. 147.
    T.E. Simos, G. Psihoyios, 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–IX (2005)Google Scholar
  148. 148.
    T.E. Simos, J. Vigo-Aguiar, 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-IX (2003)Google Scholar
  149. 149.
    T.E. Simos, I. Gutman, 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
  150. 150.
    W. Zhang, T.E. Simos, A high-order two-step phase-fitted method for the numerical solution of the Schrödinger equation. Mediterr. J. Math. 13(6), 5177–5194 (2016)CrossRefGoogle Scholar
  151. 151.
    M. Dong, T.E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat (in press) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Group of Modern Computational MethodsUral Federal UniversityEkaterinburgRussian Federation
  2. 2.Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and InformaticsUniversity of PeloponneseTripolisGreece
  3. 3.AthensGreece

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