Journal of Mathematical Chemistry

, Volume 55, Issue 3, pp 697–716 | Cite as

A new fourteenth algebraic order finite difference method for the approximate solution of the Schrödinger equation

Original Paper
First Online:
Received:
Accepted:

Abstract

In the present paper we will develop and analyse a new five-stages symmetric two-step method of high algebraic order with vanished phase-lag and its first, second, third, fourth and fifth derivatives. We will construct the new method. We will compute its local local truncation error (LTE). We will produce the asymptotic form of the LTE applying the new method to the radial time independent Schrödinger equation and we will compare it with other asymptotic forms of LTE of similar methods. Applying the new method to a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, we will investigate the stability and the interval of periodicity of the new method based and we will compare the produced interval of periodicity with other intervals of similar methods. Finally, we will examine the effectiveness of the new method applying it to the coupled Schrödinger equations.

Keywords

Phase-lag Derivative of the phase-lag Initial value problems Oscillating solution Symmetric Hybrid Multistep Hybrid Schrödinger equation 

Mathematics Subject Classification

65L05 
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Notes

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest

Supplementary material

10910_2016_704_MOESM1_ESM.pdf (127 kb)
Supplementary material 1 (pdf 126 KB)

References

  1. 1.
    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, 3880–3889 (2012)CrossRefGoogle Scholar
  2. 2.
    A.D. Raptis, T.E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problem. BIT 31, 160–168 (1991)CrossRefGoogle Scholar
  3. 3.
    D.G. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100(5), 1694–1700 (1990)CrossRefGoogle Scholar
  4. 4.
    J.M. Franco, M. Palacios, J. Comput. Appl. Math. 30, 1 (1990)CrossRefGoogle Scholar
  5. 5.
    J.D. Lambert, Numerical Methods for Ordinary Differential Systems, the Initial Value Problem (Wiley, New York, 1991)Google Scholar
  6. 6.
    E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)CrossRefGoogle Scholar
  7. 7.
    G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, 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)Google Scholar
  8. 8.
    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
  9. 9.
  10. 10.
    T.E. Simos, P.S. Williams, Bessel and Neumann fitted methods for the numerical solution of the radial Schrödinger equation. Comput. Chem. 21, 175–179 (1977)CrossRefGoogle Scholar
  11. 11.
    T.E. Simos, J. Vigo-Aguiar, A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems. Comput. Phys. Commun. 152, 274–294 (2003)CrossRefGoogle Scholar
  12. 12.
    T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175(1), 137–147 (2005)CrossRefGoogle Scholar
  13. 13.
    T. Lyche, Chebyshevian multistep methods for ordinary differential equations. Numer. Math. 19, 65–75 (1972)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)CrossRefGoogle Scholar
  16. 16.
    J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)CrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    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
  19. 19.
    Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math. 58(1), 75–82 (2003)CrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    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
  22. 22.
    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
  23. 23.
    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
  24. 24.
    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
  25. 25.
    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
  26. 26.
    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
  27. 27.
    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
  28. 28.
    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
  29. 29.
    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
  30. 30.
    T.E. Simos, New stable closed Newton-Cotes trigonometrically fitted formulae for long-time integration. Abstr. Appl. Anal. (2012). doi: 10.1155/2012/182536 Google Scholar
  31. 31.
    T.E. Simos, Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag. J. Appl. Math. (2012). doi: 10.1155/2012/420387 Google Scholar
  32. 32.
    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, 3880–3889 (2012)CrossRefGoogle Scholar
  33. 33.
    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
  34. 34.
    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(8), 1808–1834 (2015)CrossRefGoogle Scholar
  35. 35.
    I. Alolyan, T.E. Simos, A high algebraic order predictorcorrector 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
  36. 36.
    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
  37. 37.
    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
  38. 38.
    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(3), 917–947 (2014)CrossRefGoogle Scholar
  39. 39.
    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)Google Scholar
  40. 40.
    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
  41. 41.
    G.A. Panopoulos, T.E. Simos, A new optimized symmetric 8-step semi-embedded predictor–corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems. J. Math. Chem. 51(7), 1914–1937 (2013)CrossRefGoogle Scholar
  42. 42.
    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. Construction and theoretical analysis. J. Math. Chem. 51(1), 194–226 (2013)CrossRefGoogle Scholar
  43. 43.
    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
  44. 44.
    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
  45. 45.
    Th Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7(1), 81–85 (2013)CrossRefGoogle Scholar
  46. 46.
    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
  47. 47.
    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. Article Number: 910624 (2013)Google Scholar
  48. 48.
    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
  49. 49.
    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
  50. 50.
    Ch. Tsitouras, I.Th Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)Google Scholar
  51. 51.
    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
  52. 52.
    Z. Kalogiratou, Th Monovasilis, T.E. Simos, New modified Runge–Kutta–Nystrom methods for the numerical integration of the Schrödinger equation. Comput. Math. Appl. 60(6), 1639–1647 (2010)CrossRefGoogle Scholar
  53. 53.
    Th 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
  54. 54.
    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
  55. 55.
    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
  56. 56.
    Z. Kalogiratou, Th. 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)Google Scholar
  57. 57.
    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)Google Scholar
  58. 58.
    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
  59. 59.
    A. Konguetsof, T.E. Simos, An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems. Comput. Math. Appl. 45(1–3), 547–554 (2003)CrossRefGoogle Scholar
  60. 60.
    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
  61. 61.
    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
  62. 62.
    G.A. Panopoulos, T.E. Simos, An eight-step semi-embedded predictorcorrector 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
  63. 63.
    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
  64. 64.
    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
  65. 65.
    L.G. Ixaru, M. Micu, Topics in Theoretical Physics, Central Institute of Physics (Central Institute of Physics, Bucharest, 1978)Google Scholar
  66. 66.
    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
  67. 67.
    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
  68. 68.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
  69. 69.
    G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
  70. 70.
    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
  71. 71.
    M.M. Chawla, P.S. Rao, An Noumerov-typ method with minimal phase-lag for the integration of second order periodic initial-value problems II explicit method. J. Comput. Appl. Math. 15, 329–337 (1986)CrossRefGoogle Scholar
  72. 72.
    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
  73. 73.
    T.E. Simos, A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46, 981–1007 (2009)CrossRefGoogle Scholar
  74. 74.
    A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224252 (2010)CrossRefGoogle Scholar
  75. 75.
    A.D. Raptis, J.R. Cash, A variable step method for the numerical integration of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 36, 113–119 (1985)CrossRefGoogle Scholar
  76. 76.
    A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)CrossRefGoogle Scholar
  77. 77.
    R.B. Bernstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules. Proc. R. Soc. Ser. A 274, 427–442 (1963)CrossRefGoogle Scholar
  78. 78.
    R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)CrossRefGoogle Scholar
  79. 79.
    T.E. Simos, Exponentially fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation and related problems. Comput. Mater. Sci. 18, 315–332 (2000)CrossRefGoogle Scholar
  80. 80.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
  81. 81.
    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, 1239–1256 (2015)CrossRefGoogle Scholar
  82. 82.
    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
  83. 83.
    X. Xi, T.E. Simos, A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 54(7), 1417–1439 (2016)CrossRefGoogle Scholar
  84. 84.
    F. Hui, T.E. Simos, Hybrid high algebraic order two-step method with vanished phase-lag and its first and second derivatives. MATCH Commun. Math. Comput. Chem. 73, 619–648 (2015)Google Scholar
  85. 85.
    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, 442–465 (2016)CrossRefGoogle Scholar
  86. 86.
    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
  87. 87.
    L. Zhang, T.E. Simos, An efficient numerical method for the solution of the Schrödinger equation. Adv. Math. Phys. (in press)Google Scholar
  88. 88.
    M. Dong, T.E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat (to appear)Google Scholar
  89. 89.
    R. Lin, T.E. Simos, A two-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Open Phys. (to appear)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of Information EngineeringChang’an UniversityXi’anPeople’s Republic of China
  2. 2.Group of Modern Computational MethodsUral Federal UniversityEkaterinburgRussian Federation
  3. 3.Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and InformaticsUniversity of PeloponneseTripolisGreece
  4. 4.AthensGreece

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