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Mulitstep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation

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Abstract

A tenth algebraic order eight-step method is developed in this paper. For this method  we require the phase-lag and its first and second derivatives to be vanished. A comparative error analysis and a comparative stability analysis are also presented in this paper. The new proposed method is applied for the numerical solution of the one-dimensional Schrödinger equation. The efficiency of the new methodology is proved via the theoretical analysis and the numerical applications. General conclusions about the importance of several properties on the construction of numerical algorithms for the approximate solution of the radial Schrödinger equation are also presented.

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References

  1. Ixaru L.G., Micu M.: Topics in Theoretical Physics. Central Institute of Physics, Bucharest (1978)

    Google Scholar 

  2. Landau L.D., Lifshitz F.M.: Quantum Mechanics. Pergamon, New York (1965)

    Google Scholar 

  3. Prigogine, I., Rice, S. (eds): Advances in Chemical Physics vol. 93: New Methods in Computational Quantum Mechanics. John Wiley & Sons, New York (1997)

    Google Scholar 

  4. Herzberg G.: Spectra of Diatomic Molecules. Van Nostrand, Toronto (1950)

    Google Scholar 

  5. T.E. Simos, Atomic structure computations in chemical modelling: applications and theory (Editor: A. Hinchliffe, UMIST). R. Soc. Chem. 38–142 (2000)

  6. Simos T.E.: Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems, chemical modelling: application and theory. R. Soc. Chem. 2, 170–270 (2002)

    CAS  Google Scholar 

  7. T.E. Simos, Numerical Solution of Ordinary Differential Equations with Periodical Solution. Doctoral Dissertation, National Technical University of Athens, Greece, 1990 (in Greek)

  8. 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)

    Article  Google Scholar 

  9. Dormand J.R., Prince P.J.: A family of embedded RungeKutta formulae. J. Comput. Appl. Math. 6, 1926 (1980)

    Article  Google Scholar 

  10. Sideridis A.B., Simos T.E.: A low-order embedded Runge-Kutta method for periodic initial-value problems. J. Comput. Appl. Math. 44(2), 235–244 (1992)

    Article  Google Scholar 

  11. Simos T.E.: A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial value problems with oscillating solution. Comput. Math. Appl. 25, 95–101 (1993)

    Article  Google Scholar 

  12. Simos T.E.: Runge-Kutta interpolants with minimal phase-lag. Comput. Math. Appl. 26, 43–49 (1993)

    Article  Google Scholar 

  13. Simos T.E.: Runge-Kutta-Nyström interpolants for the numerical integration of special second-order periodic initial-value problems. Comput. Math. Appl. 26, 7–15 (1993)

    Article  Google Scholar 

  14. Simos T.E.: A high-order predictor-corrector method for periodic IVPs. Appl. Math. Lett. 6(5), 9–12 (1993)

    Article  Google Scholar 

  15. Simos T.E., Dimas E., Sideridis A.B.: A Runge-Kutta-Nyström method for the numerical-integration of special 2nd-order periodic initial-value problems. J. Comput. Appl. Math. 51(3), 317–326 (1994)

    Article  Google Scholar 

  16. Simos T.E.: An explicit high-order predictor-corrector method for periodic initial-value problems. Math. Models Methods Appl. Sci. 5(2), 159–166 (1995)

    Article  Google Scholar 

  17. Avdelas G., Simos T.E.: Block Runge-Kutta methods for periodic initial-value problems. Comput. Math. Appl. 31, 69–83 (1996)

    Article  Google Scholar 

  18. Avdelas G., Simos T.E.: Embedded methods for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 31, 85–102 (1996)

    Article  Google Scholar 

  19. Simos T.E.: A modified Runge-Kutta method for the numerical solution of ODE’s with oscillation solutions. Appl. Math. Lett. 9(6), 61–66 (1996)

    Article  Google Scholar 

  20. Simos T.E.: Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations. J. Math. Chem. 24(1–3), 23–37 (1998)

    Article  CAS  Google Scholar 

  21. Simos T.E.: An embedded Runge-Kutta method with phase-lag of order infinity for the numerical solution of the of Schrödinger equation. Int. J. Mod. Phys. C 11, 1115–1133 (2000)

    Article  Google Scholar 

  22. Simos T.E., Vigo-Aguiar J.: A new modified Runge-Kutta-Nyström method with phase-lag of order infinity for the numerical solution of the Schrödinger equation and related problems. Int. J. Mod. Phys. C 11, 1195–1208 (2000)

    Article  Google Scholar 

  23. Simos T.E., Vigo-Aguiar J.: A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the of Schrödinger equation and related problems. Comput. Chem. 25, 275–281 (2001)

    Article  CAS  Google Scholar 

  24. 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)

    Article  CAS  Google Scholar 

  25. Simos T.E., Williams P.S.: A New Runge-Kutta-Nyström method with phase-lag of order infinity for the numerical solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 45, 123–137 (2002)

    CAS  Google Scholar 

  26. Tsitouras C., Simos T.E.: Optimized Runge-Kutta pairs for problems with oscillating solutions. J. Comput. Appl. Math. 147(2), 397–409 (2002)

    Article  Google Scholar 

  27. Anastassi Z.A., Simos T.E.: Special optimized Runge-Kutta methods for IVPs with oscillating solutions. Int. J. Mod. Phys. C 15, 1–15 (2004)

    Article  Google Scholar 

  28. Anastassi Z.A., Simos T.E.: A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for the numerical solution of orbital problems. New Astron. 10, 31–37 (2004)

    Article  Google Scholar 

  29. 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)

    Article  Google Scholar 

  30. 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)

    Article  Google Scholar 

  31. Triantafyllidis T.V., Anastassi Z.A., Simos T.E.: Two optimized Runge-Kutta methods for the solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 753–771 (2008)

    CAS  Google Scholar 

  32. Papadopoulos D.F., Anastassi Z.A., Simos T.E.: A phase-fitted Runge-Kutta-Nystrom method for the numerical solution of initial value problems with oscillating solutions. Comput. Phys. Commun. 180(10), 1839–1846 (2009)

    Article  CAS  Google Scholar 

  33. 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)

    Article  CAS  Google Scholar 

  34. 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)

    Article  CAS  Google Scholar 

  35. 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)

    Article  CAS  Google Scholar 

  36. 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)

    Article  CAS  Google Scholar 

  37. Lambert J.D., Watson I.A.: Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)

    Article  Google Scholar 

  38. Raptis A.D., Allison A.C.: Exponential—fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)

    Article  Google Scholar 

  39. Raptis A.D.: Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control. Comput. Phys. Commun. 28, 427–431 (1983)

    Article  Google Scholar 

  40. Kalogiratou Z., Simos T.E.: A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation. Appl. Math. Comput. 112, 99–112 (2000)

    Article  Google Scholar 

  41. Raptis A.D., Simos T.E.: A four-step phase-fitted method for the numerical integration of second order initial-value problem. BIT 31, 160–168 (1991)

    Article  Google Scholar 

  42. Simos T.E., Raptis A.D.: Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation. Computing 45, 175–181 (1990)

    Article  Google Scholar 

  43. Simos T.E.: A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems. Int. J. Comput. Math. 39, 135–140 (1991)

    Article  Google Scholar 

  44. Simos T.E.: A Numerov-type method for the numerical-solution of the radial Schrödinger-equation. Appl. Numer. Math. 7(2), 201–206 (1991)

    Article  Google Scholar 

  45. Simos T.E.: Explicit two-step methods with minimal phase-lag for the numerical-integration of special second-order initial-value problems and their application to the one-dimensional Schrödinger-equation. J. Comput. Appl. Math. 39(1), 89–94 (1992)

    Article  Google Scholar 

  46. Simos T.E.: Two-step almost P-stable complete in phase methods for the numerical integration of second order periodic initial-value problems. Inter. J. Comput. Math. 46, 77–85 (1992)

    Article  Google Scholar 

  47. Simos T.E.: An explicit almost P-stable two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems. Appl. Math. Comput. 49, 261–268 (1992)

    Article  Google Scholar 

  48. Simos T.E.: High—order methods with minimal phase-lag for the numerical integration of the special second order initial value problem and their application to the one-dimensional Schrödinger equation. Comput. Phys. Commun. 74, 63–66 (1993)

    Article  Google Scholar 

  49. Simos T.E.: A new variable-step method for the numerical-integration of special 2Nd-order initial-value problems and their application to the one-dimensional SchrÖdinger-equation. Appl. Math. Lett. 6(3), 67–73 (1993)

    Article  Google Scholar 

  50. Simos T.E.: A family of two-step almost P-stable methods with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems. Jpn. J. Ind. Appl. Math. 10, 289–297 (1993)

    Article  Google Scholar 

  51. Simos T.E.: A predictor-corrector phase-fitted method for y′′ = f(x,y). Math. Comput. Simul. 35, 153–159 (1993)

    Article  Google Scholar 

  52. Simos T.E.: A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial value problems. Proc. R. Soc. Lond. A 441, 283–289 (1993)

    Article  Google Scholar 

  53. Simos T.E.: An explicit 4-step phase-fitted method for the numerical-integration of 2nd-order initial-value problems. J. Comput. Appl. Math. 55(2), 125–133 (1994)

    Article  Google Scholar 

  54. Simos T.E.: Some new variable-step methods with minimal phase-lag for the numerical integration of special 2nd-order initial value problems. Appl. Math. Comput. 64, 65–72 (1994)

    Article  Google Scholar 

  55. Simos T.E., Mousadis G.: Some new Numerov-type methods with minimal phase-lag for the numerical integration of the radial Schrödinger equation. Mol. Phys. 83, 1145–1153 (1994)

    Article  CAS  Google Scholar 

  56. Simos T.E., Mousadis G.: A two-step method for the numerical solution of the radial Schrdinger equation. Comput. Math. Appl. 29, 31–37 (1995)

    Article  Google Scholar 

  57. Simos T.E.: Predictor corrector phase-fitted methods for Y′′ = F(X,Y) and an application to the Schrödinger-equation. Int. J. Quantum Chem. 53(5), 473–483 (1995)

    Article  CAS  Google Scholar 

  58. Simos T.E.: Some low-order two-step almost P-stable methods with phase-lag of order infinity for the numerical integration of the radial Schrödinger equation. Int. J. Modern Phys. A 10, 2431–2438 (1995)

    Article  Google Scholar 

  59. Simos T.E.: A new Numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation. Int. J. Modern Phys. C-Phys. Comput. 7(1), 33–41 (1996)

    Article  Google Scholar 

  60. Papakaliatakis G., Simos T.E.: A new method for the numerical solution of fourth order BVPs with oscillating solutions. Comput. Math. Appl. 32, 1–6 (1996)

    Article  Google Scholar 

  61. Simos T.E.: Accurate computations for the elastic scattering phase-shift problem. Comput. Chem. 21, 125–128 (1996)

    Article  Google Scholar 

  62. Simos T.E.: An eighth order method with minimal phase-lag for accuarate computations for the elastic scattering phase-shift problem. Inter. J. Modern Phys. C 7, 825–835 (1996)

    Article  Google Scholar 

  63. Simos T.E.: Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem. J. Math. Chem. 21(4), 359–372 (1997)

    Article  CAS  Google Scholar 

  64. Simos T.E.: An extended Numerov-type method for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 33, 67–78 (1997)

    Article  Google Scholar 

  65. Simos T.E.: New Numerov-type methods for computing eigenvalues, resonances and phase shifts of the radial Schrödinger equation. Int. J. Quantum Chem. 62, 467–475 (1997)

    Article  CAS  Google Scholar 

  66. Simos T.E.: New P-stable high-order methods with minimal phase-lag for the numerical integration of the radial Schrödinger equation. Phys. Scr. 55, 644–650 (1997)

    Article  CAS  Google Scholar 

  67. Simos T.E.: Eighth order methods for elastic scattering phase-shifts. Int. J. Theor. Phys. 36, 663–672 (1997)

    Article  Google Scholar 

  68. Simos T.E., Tougelidis G.: An explicit eighth order method with minimal phase-lag for the numerical solution of the Schrödinger equation. Comput. Mater. Sci. 8, 317–326 (1997)

    Article  Google Scholar 

  69. Simos T.E., Tougelidis G.: An explicit eighth order method with minimal phase-lag for accurate computations of eigenvalues, resonances and phase shifts. Comput. Chem. 21, 327–334 (1997)

    Article  CAS  Google Scholar 

  70. Simos T.E.: Eighth order method for accurate computations for the elastic scattering phase-shift problem. Int. J. Quantum Chem. 68, 191–200 (1998)

    Article  CAS  Google Scholar 

  71. Simos T.E.: New embedded explicit methods with minimal phase-lag for the numerical integration of the Schrödinger equation. Comput. Chem. 22, 433–440 (1998)

    Article  CAS  Google Scholar 

  72. Simos T.E.: High-algebraic, high-phase-lag methods for accurate computations for the elastic-scattering phase shift problem. Can. J. Phys. 76, 473–493 (1998)

    Article  CAS  Google Scholar 

  73. Simos T.E.: High algebraic order methods with minimal phase-lag for accurate solution of the Schrödinger equation. Int. J. Modern Phys. C 9, 1055–1071 (1998)

    Article  Google Scholar 

  74. Avdelas G., Simos T.E.: Embedded eighth order methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 26(4), 327–341 (1999)

    Article  Google Scholar 

  75. Simos T.E.: A new finite difference scheme with minimal phase-lag for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 106, 245–264 (1999)

    Article  Google Scholar 

  76. Simos T.E.: High algebraic order explicit methods with reduced phase-lag for an efficient solution of the Schrödinger equation. Int. J. Quantum Chem. 73, 479–496 (1999)

    Article  CAS  Google Scholar 

  77. Simos T.E.: Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Comput. Chem. 23, 439–446 (1999)

    Article  CAS  Google Scholar 

  78. Simos T.E.: Explicit eighth order methods for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Commun. 119, 32–44 (1999)

    Article  CAS  Google Scholar 

  79. Simos T.E.: High algebraic order methods for the numerical solution of the Schrödinger equation. Mol. Simul. 22, 303–349 (1999)

    Article  CAS  Google Scholar 

  80. Avdelas G., Konguetsof A., Simos T.E.: A family of hybrid eighth order methods with minimal phase-lag for the numerical solution of the Schrödinger equation and related problems. Int. J. Modern Phys. C 11, 415–437 (2000)

    Google Scholar 

  81. Avdelas G., Simos T.E.: Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Phys. Rev. E 62, 1375–1381 (2000)

    Article  CAS  Google Scholar 

  82. Avdelas G., Simos T.E.: On variable-step methods for the numerical solution of Schrödinger equation and related problems. Comput. Chem. 25, 3–13 (2001)

    Article  CAS  Google Scholar 

  83. Simos T.E., Williams P.S.: New insights in the development of Numerov-type methods with minimal phase-lag for the numerical solution of the Schrödinger equation. Comput. Chem. 25, 77–82 (2001)

    Article  CAS  Google Scholar 

  84. Avdelas G., Konguetsof A., Simos T.E.: A generator of hybrid explicit methods for the numerical solution of the Schrödinger equation and related problems. Comput. Phys. Commun. 136, 14–28 (2001)

    Article  CAS  Google Scholar 

  85. Simos T.E., Vigo-Aguiar J.: A symmetric high-order method with minimal phase-lag for the numerical solution of the Schrödinger equation. Int. J. Modern Phys. C 12, 1035–1042 (2001)

    Article  Google Scholar 

  86. Simos T.E., Vigo-Aguiar J.: On the construction of efficient methods for second order IVPs with oscillating solution. Int. J. Modern Phys. C 12, 1453–1476 (2001)

    Article  Google Scholar 

  87. 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)

    Article  CAS  Google Scholar 

  88. 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)

    Article  CAS  Google Scholar 

  89. Tsitouras Ch., Simos T.E.: High algebraic, high phase-lag order embedded Numerov-type methods for oscillatory problems. Appl. Math. Comput. 131, 201–211 (2002)

    Article  Google Scholar 

  90. Avdelas G., Konguetsof A., Simos T.E.: A generator of dissipative methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 148, 59–73 (2002)

    Article  CAS  Google Scholar 

  91. Konguetsof A., Simos T.E.: P-stable eighth algebraic order methods for the numerical solution of the Schrödinger equation. Comput. Chem. 26, 105–111 (2002)

    Article  CAS  Google Scholar 

  92. 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)

    Article  CAS  Google Scholar 

  93. 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)

    Article  Google Scholar 

  94. Simos T.E., Famelis I.T., Tsitouras Ch.: Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27–40 (2003)

    Article  Google Scholar 

  95. 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)

    Article  Google Scholar 

  96. 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)

    Article  CAS  Google Scholar 

  97. Panopoulos G.A., Anastassi Z.A., Simos T.E.: Two new optimized eight-step symmetric methods for the efficient solution of the Schrödinger equation and related problems. MATCH Commun. Math. Comput. Chem. 60(3), 773–785 (2008)

    CAS  Google Scholar 

  98. Simos T.E.: A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46(3), 981–1007 (2009)

    Article  CAS  Google Scholar 

  99. Simos T.E.: A family of 4-step exponentially fitted predictor-corrector methods for the numerical-integration of the Schrödinger-equation. J. Comput. Appl. Math. 58(3), 337–344 (1995)

    Article  Google Scholar 

  100. Thomas R.M., Simos T.E., Mitsou G.V.: A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation. J. Comput. Appl. Math. 67(2), 255–270 (1996)

    Article  Google Scholar 

  101. Thomas R.M., Simos T.E.: A family of hybrid exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation. J. Comput. Appl. Math. 87(2), 215–226 (1997)

    Article  Google Scholar 

  102. Psihoyios G., Simos T.E.: Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math. 158(1), 135–144 (2003)

    Article  Google Scholar 

  103. Konguetsof A.: A new two-step hybrid method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871–890 (2010)

    Article  CAS  Google Scholar 

  104. 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)

    Article  CAS  Google Scholar 

  105. 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)

    Article  CAS  Google Scholar 

  106. 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)

    Article  CAS  Google Scholar 

  107. 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)

    Article  CAS  Google Scholar 

  108. 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)

    Article  CAS  Google Scholar 

  109. Kalogiratou Z., Simos T.E.: Newton-Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)

    Article  Google Scholar 

  110. 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)

    Article  Google Scholar 

  111. 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)

    CAS  Google Scholar 

  112. 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)

    Article  CAS  Google Scholar 

  113. 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)

    Article  CAS  Google Scholar 

  114. Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae for numerical integration of the Schrödinger equation. Comput. Lett. 3(1), 45–57 (2007)

    Article  Google Scholar 

  115. Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Revista Mexicana de Astronomia y Astrofysica 42(2), 167–177 (2006)

    Google Scholar 

  116. Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae for long-time integration. Int. J. Modern Phys. C 14(8), 1061–1074 (2003)

    Article  Google Scholar 

  117. 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)

    Article  Google Scholar 

  118. 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)

    Article  Google Scholar 

  119. Simos T.E.: A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 25(1), 65–84 (1999)

    Article  CAS  Google Scholar 

  120. 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)

    Article  CAS  Google Scholar 

  121. 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)

    Article  CAS  Google Scholar 

  122. 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)

    Article  CAS  Google Scholar 

  123. Simos T.E.: Exponentially—fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)

    Article  CAS  Google Scholar 

  124. 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)

    Article  CAS  Google Scholar 

  125. Simos T.E.: A family of four-step trigonometrically-fitted methods and its application to the Schrödinger equation. J. Math. Chem. 44(2), 447–466 (2009)

    Article  CAS  Google Scholar 

  126. Simos T.E.: Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Appl. Math. 110(3), 1331–1352 (2010)

    Article  Google Scholar 

  127. 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)

    Article  CAS  Google Scholar 

  128. 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)

    Article  CAS  Google Scholar 

  129. 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)

    Article  CAS  Google Scholar 

  130. 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)

    Article  CAS  Google Scholar 

  131. 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)

    Article  CAS  Google Scholar 

  132. Simos T.E.: Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004)

    Article  Google Scholar 

  133. Simos T.E., Williams P.S.: On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513–554 (1999)

    Article  CAS  Google Scholar 

  134. Anastassi Z.A., Simos T.E.: Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Rep. Rev. Sect. Phys. Lett. 482, 1–240 (2009)

    Google Scholar 

  135. 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)

    Article  CAS  Google Scholar 

  136. 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)

    Article  CAS  Google Scholar 

  137. 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), 165–165 (2004)

    Article  Google Scholar 

  138. Simos T.E., 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–IX (2005)

    Article  Google Scholar 

  139. Simos T.E., 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–IX (2003)

    Article  Google Scholar 

  140. Simos T.E., Williams P.S.: A finite-difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79(2), 189–205 (1997)

    Article  Google Scholar 

  141. 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)

    Article  CAS  Google Scholar 

  142. 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)

    Article  Google Scholar 

  143. Quinlan G.D., Tremaine S.: Symmetric multistep methods for the numerical integration of planetary orbits. astron. J. 100(5), 1694–1700 (1990)

    Article  Google Scholar 

  144. I. Alolyan, T.E. Simos, High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation. J. Math. Chem. (2010). doi:10.1007/s10910-010-9718-y

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia

    Ibraheem Alolyan & T. E. Simos

  2. Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, 221 00, Tripolis, Greece

    T. E. Simos

  3. 10 Konitsis Street, Amfithea—Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

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  1. Ibraheem Alolyan
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  2. T. E. Simos
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Correspondence to T. E. Simos.

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Highly Cited Researcher, Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences Corresponding Member of European Academy of Arts, Sciences and Humanities.

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Alolyan, I., Simos, T.E. Mulitstep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation. J Math Chem 48, 1092–1143 (2010). https://doi.org/10.1007/s10910-010-9728-9

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