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

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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.

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References

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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

    Article  Google Scholar 

  7. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7(1), 81–85 (2013)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  21. C. Tsitouras, I.T. Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)

    Article  Google Scholar 

  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)

  23. http://burtleburtle.net/bob/math/multistep.html

  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)

    Article  CAS  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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. 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. 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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  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. 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)

    CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  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. T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae for long-time integration. Int. J. Mod. Phys. C 14(8), 061–1074 (2003)

    Article  Google Scholar 

  92. T.E. Simos, New closed Newton–Cotes type formulae as multilayer symplectic integrators. J. Chem. Phys. 133(10), Article Number: 104108 (2010)

  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. 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)

    Article  CAS  Google Scholar 

  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. T.E. Simos, New open modified Newton Cotes type formulae as multilayer symplectic integrators. Appl. Math. Model. 37(4), 1983–1991 (2013)

    Article  Google Scholar 

  97. G. Vanden Berghe, M. Van Daele, Exponentially fitted open Newton–Cotes differential methods as multilayer symplectic integrators. J. Chem. Phys. 132, 204107 (2010)

  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)

  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)

    Article  CAS  Google Scholar 

  100. T. Monovasilis, T.E. Simos, Symplectic methods for the numerical integration of the Schrödinger equation. Comput. Mater. Sci. 38(3), 526–532 (2007)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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. 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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  125. Z. Wang, P-stable linear symmetric multistep methods for periodic initial-value problems. Comput. Phys. Commun. 171(3), 162–174 (2005)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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. S. Kottwitz, LaTeX Cookbook, pp. 231–236 (Packt Publishing Ltd., Birmingham B3 2PB, UK 2015)

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

    Google Scholar 

  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)

    Article  Google Scholar 

  142. J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  Google Scholar 

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

    Google Scholar 

  144. I. Prigogine (Eds): Advances in Chemical Physics Vol. 93: New Methods in Stuart Rice Computational Quantum Mechanics (Wiley, 1997)

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

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  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. 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)

    Article  Google Scholar 

  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)

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  1. Group of Modern Computational Methods, Ural Federal University, 19 Mira Street, Ekaterinburg, Russian Federation, 620002

    Dmitriy B. Berg & T. E. Simos

  2. Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00, Tripolis, Greece

    T. E. Simos

  3. 10 Konitsis Street, Amphithea - Paleon Phaleron, 175 64, Athens, Greece

    T. E. Simos

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  1. Dmitriy B. Berg
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T. E. Simos: Highly Cited Researcher (http://isihighlycited.com/), 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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Berg, D.B., Simos, T.E. An efficient six-step method for the solution of the Schrödinger equation. J Math Chem 55, 1521–1547 (2017). https://doi.org/10.1007/s10910-017-0742-z

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