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A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation

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Abstract

In this paper a four stages symmetric two-step method with vanished phase-lag and its first derivative with high algebraic order is developed for the first time in the literature. More pricesly in this paper the following are presented: (1) the phase-lag analysis for the construction of the new high algebraic order method, (2) the construction of the new method, (3) the error analysis based on the radial Schrödinger equation, (4) the interval of periodicity analysis and the stability analysis, (5) a new error estimation procedure based on the phase-lag (6) the numerical tests for the study of the efficiency of the new obtained method which are based on the numerical solution of the Schrödinger equation.

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Notes

  1. Where S is a set of distinct points.

References

  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)

    Article  Google Scholar 

  2. J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189202 (1976)

    Article  Google Scholar 

  3. T.E. Simos, P.S. Williams, A finite-difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189205 (1997)

    Article  Google Scholar 

  4. R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225238 (1984)

    Article  Google Scholar 

  5. 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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. J.M. Franco, M. Palacios, J. Comput. Appl. Math. 30, 1 (1990)

    Article  Google Scholar 

  8. J.D. Lambert, Numerical methods for ordinary differential systems, the initial value problem (Wiley, New York, 1991)

    Google Scholar 

  9. E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)

    Article  Google Scholar 

  10. 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)

    CAS  Google Scholar 

  11. 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 

  12. http://www.burtleburtle.net/bob/math/multistep.html

  13. 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)

    Article  Google Scholar 

  14. T.E. Simos, Jesus 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)

    Article  CAS  Google Scholar 

  15. T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175 (1): 137–147 (2005)

  16. T. Lyche, Chebyshevian multistep methods for ordinary differential equations. Num. Math. 19, 65–75 (1972)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  22. 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 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  29. 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 

  30. T. E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration, abstract and applied analysis, Volume 2012, Article ID 182536, 15 pp. (2012). doi:10.1155/2012/182536

  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. Volume 2012, Article ID 420387, 17 pp. (2012). doi:10.1155/2012/420387

  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, 38803889 (2012)

    Article  Google Scholar 

  33. Ibraheem 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 

  34. Ibraheem 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)

    Article  CAS  Google Scholar 

  35. Ibraheem Alolyan, T.E. Simos, A high algebraic orderpredictor–corrector explicit method with vanished phase-lag and itsfirst, second, third and fourth derivatives for the numericalsolution of the Schrödinger equation and related problems. J. Math. Chem. 53(7), 1495–1522 (2015)

    Article  CAS  Google Scholar 

  36. Ibraheem 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 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  40. Ibraheem 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 

  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)

    Article  CAS  Google Scholar 

  42. T.E. Simos, Part I: construction and theoretical analysis. New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrdinger equation. J. Math. Chem. 51(1), 194–226 (2013)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  44. Dimitris 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 

  45. Th 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 

  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)

    Article  Google Scholar 

  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. (2013). doi:10.1155/2013/910624

  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)

    Article  Google Scholar 

  49. Ibraheem 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 

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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  52. Z. Kalogiratou, Th 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  54. T. Monovasilis, Z. Kalogiratou and T. E. Simos, Construction of exponentially fitted symplectic Runge–Kutta–Nyström methods from partitioned Runge–Kutta methods. Mediterr. J. Math. (in press)

  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. Higinio Ramos, Z. Kalogiratou, Th. Monovasilis, T.E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms (2015). doi:10.1007/s11075-015-0081-8

  57. 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)

    Article  Google Scholar 

  58. 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)

    Article  Google Scholar 

  59. 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 

  60. 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 

  61. 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 

  62. Fei 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, 2191–2213 (2015)

  63. L.Gr 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 

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

    Google Scholar 

  65. 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 

  66. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  69. 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 

  70. 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 

  71. 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 

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

    Article  CAS  Google Scholar 

  73. A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224252 (2010)

    Article  Google Scholar 

  74. Kenan 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)

    Article  CAS  Google Scholar 

  75. A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)

    Article  Google Scholar 

  76. 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)

    Article  CAS  Google Scholar 

  77. R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)

    Article  CAS  Google Scholar 

  78. 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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Information Engineering, Changan University, Xi’an, 710064, People’s Republic of China

    Minjian Liang

  2. Guangdong Special Equipment Inspection and Research Institute Branch of Zhuhai, Zhuhai, 519002, People’s Republic of China

    Minjian Liang

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

    T. E. Simos

  4. 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

  5. 10 Konitsis Street, Amfithea - Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

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  1. Minjian Liang
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T. E. Simos is a 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.

Appendices

Appendix 1: Formulae of the derivatives of \(q_{n}\)

Formulae of the derivatives which presented in the formulae of the Local Truncation Errors:

$$\begin{aligned} q_{n}^{(2)}= & {} \left( V(x)-V_{c} + G\right) \, q(x) \\ q_{n}^{(3)}= & {} \left( {\frac{d}{dx}}g \left( x \right) \right) q \left( x \right) \\&+ \left( g \left( x \right) +G \right) {\frac{d}{dx}}q \left( x \right) \\ q_{n}^{(4)}= & {} \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) q \left( x \right) \\&+2\, \left( {\frac{d}{dx}}g \left( x \right) \right) {\frac{d}{dx}}q \left( x \right) \\&+ \left( g \left( x \right) +G \right) ^{2}q \left( x \right) \\ q_{n}^{(5)}= & {} \left( {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \right) q \left( x \right) \\&+3\, \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) {\frac{d}{dx}}q \left( x \right) \\&+4\, \left( g \left( x \right) +G \right) q \left( x \right) {\frac{d}{dx}}g \left( x \right) \\&+ \left( g \left( x \right) +G \right) ^{2}{\frac{d}{dx}}q \left( x \right) \\ q_{n}^{(6)}= & {} \left( {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \right) q \left( x \right) \\&+4\, \left( {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \right) {\frac{d}{dx}}q \left( x \right) \\&+7\, \left( g \left( x \right) +G \right) q \left( x \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+4\, \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}q \left( x \right) \\&+6\, \left( g \left( x \right) +G \right) \left( {\frac{d}{dx}}q \left( x \right) \right) {\frac{d}{dx}}g \left( x \right) \\&+ \left( g \left( x \right) +G \right) ^{3}q \left( x \right) \\ q_{n}^{(7)}= & {} \left( {\frac{d^{5}}{d{x}^{5}}}g \left( x \right) \right) q \left( x \right) \\&+5\, \left( {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \right) {\frac{d}{dx}}q \left( x \right) \end{aligned}$$
$$\begin{aligned}&+11\, \left( g \left( x \right) +G \right) q \left( x \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \\&+15\, \left( {\frac{d}{dx}}g \left( x \right) \right) q \left( x \right) \\&{\frac{d^{2}}{d{x}^{2}}}g \left( x \right) +13\, \left( g \left( x \right) +G \right) \\&\left( {\frac{d}{dx}}q \left( x \right) \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+10\, \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}{\frac{d}{dx}}q \left( x \right) \\&+9\, \left( g \left( x \right) +G \right) ^{2}q \left( x \right) \\&{\frac{d}{dx}}g \left( x \right) + \left( g \left( x \right) +G \right) ^{3}{\frac{d}{dx}}q \left( x \right) \\ q_{n}^{(8)}= & {} \left( {\frac{d^{6}}{d{x}^{6}}}g \left( x \right) \right) q \left( x \right) \\&+6\, \left( {\frac{d^{5}}{d{x}^{5}}}g \left( x \right) \right) {\frac{d}{dx}}q \left( x \right) \\&+16\, \left( g \left( x \right) +G \right) q \left( x \right) {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \\&+26\, \left( {\frac{d}{dx}}g \left( x \right) \right) q \left( x \right) \\&{\frac{d^{3}}{d{x}^{3}}}g \left( x \right) +24\, \left( g \left( x \right) +G \right) \\&\left( {\frac{d}{dx}}q \left( x \right) \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \\&+15\, \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) ^{2}q \left( x \right) \end{aligned}$$
$$\begin{aligned}&+48\, \left( {\frac{d}{dx}}g \left( x \right) \right) \\&\left( {\frac{d}{dx}}q \left( x \right) \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+22\, \left( g \left( x \right) +G \right) ^{2}q \left( x \right) \\&{\frac{d^{2}}{d{x}^{2}}}g \left( x \right) +28\, \left( g \left( x \right) +G \right) \\&q \left( x \right) \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}\\&+12\, \left( g \left( x \right) +G \right) ^{2} \\&\left( {\frac{d}{dx}}q \left( x \right) \right) {\frac{d}{dx}}g \left( x \right) \\&+ \left( g \left( x \right) +G \right) ^{4}q \left( x \right) \\&\ldots \end{aligned}$$

Appendix 2: The new four stages twelfth algebraic order symmetric two-step method with vanished phase-lag (phase-fitted)

We consider the family of four stages symmetric two-step methods (12) with

$$\begin{aligned} a_{0} = -\frac{59}{113400}, \, a_{1} = \frac{15}{112}, \, a_{2}=-2, \, b_{2} = -\frac{7}{50}, \, b_{{1}}\, = \, {\frac{71}{600}} \end{aligned}$$

Applying the method (12) with coefficients \(a_{j}, \, j=0(1)2, \, \, b_{i}, \, i=1,2\) given above to the scalar test equation (4), we obtain the difference equation given by (5) with:

$$\begin{aligned} A_{1} (v)= & {} 1+{v}^{2} \left[ {\frac{229}{7788}}-{\frac{140\,{v}^{2}}{649} \left( -{\frac{127\,{v}^{2}}{16934400}}-{\frac{11}{5600}} \right) }+{\frac{127\,{v}^{4}}{78503040}} \right] \\ A_{0} (v)= & {} -2+{v}^{2} \left[ b_{{0}}-{\frac{140}{649}}-{\frac{280\,{v}^{2}}{649} \left( {\frac{711}{5600}}-{\frac{2923\,{v}^{2}}{1693440}} \right) } \right] \end{aligned}$$

In order the above four stages method (12) to have vanished the phase-lag (phase-fitted) we have the following equation:

$$\begin{aligned} \mathrm{Phase-Lag(PL)}= & {} \frac{1}{2}\,\frac{T_{6}}{1+{v}^{2} \left( {\frac{229}{7788}}-{\frac{140\,{v}^{2}}{649} \left( -{\frac{127\,{v}^{2}}{16934400}}-{\frac{11}{5600}} \right) }+{\frac{127\,{v}^{4}}{78503040}} \right) } = 0 \end{aligned}$$

where

$$\begin{aligned} T_{6}= & {} 2\, \Bigl ( 1+{v}^{2} \Bigl ( {\frac{229}{7788}}-{\frac{140\,{v}^{2}}{649} \left( -{\frac{127\,{v}^{2}}{16934400}}-{\frac{11}{5600}} \right) }\\&+{\frac{127\,{v}^{4}}{78503040}} \Bigr ) \Bigr ) \cos \left( v \right) -2+{v}^{2} \left( b_{{0}}-{\frac{140}{649}}-{\frac{280\,{v}^{2}}{649} \left( {\frac{711}{5600}}-{\frac{2923\,{v}^{2}}{1693440}} \right) } \right) \end{aligned}$$

If we solve the above equation, we obtain the coefficient \(b_{0}\) of the new developed four stages symmetric two-step method:

$$\begin{aligned} b_{{0}}\, = \,-\frac{T_{7}}{19625760\,{v}^{2}} \end{aligned}$$

where

$$\begin{aligned} T_{7}= & {} 127\,\cos \left( v \right) {v}^{6}+14615\,{v}^{6}+16632\,\cos \left( v \right) {v}^{4}\\&-1075032\,{v}^{4}+1154160\,\cos \left( v \right) {v}^{2}\\&-4233600\,{v}^{2}+39251520\,\cos \left( v \right) -39251520 \end{aligned}$$

For cancellations purposes we give also the Taylor series expansion :

$$\begin{aligned} b_{{0}}= & {} {\frac{4505}{3894}}-{\frac{45469\,{v}^{12}}{1697361329664000}}+{\frac{12079\,{v}^{14}}{20368335955968000}} \\&-{\frac{26123\,{v}^{16}}{4155140535017472000}}+{\frac{697769\,{v}^{18}}{16579010734719713280000}} + \ldots \end{aligned}$$

The behavior of the coefficient of the new four stages two-step method is presented in Fig. 3.

Fig. 3
figure 3

Behavior of the coefficient of the new obtained twelfth algebraic order four stages two-step method for several values of \(v = \phi \, h\)

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Based on the above coefficients, we can find the local truncation error of the new developed two stage two-step method (12) (mentioned as FourStageTwoStep100D) which is given by:

$$\begin{aligned}&LTE _{FourStageTwoStep120D}= \\&\quad - \frac{45469}{1697361329664000}\, h^{14} \, \Biggl ( q_{n}^{(14)} - \phi ^{12} \, q_{n}^{(2)} \Biggr ) + O \left( h^{14} \right) \end{aligned}$$

The asymptotic form of the Local Truncation Error (after application to the test problem (20)) is given by:

$$\begin{aligned}&LTE _{FourStageTwoStep120D}= \\&\quad - \frac{45469}{282893554944000}\, h^{14} \, \Biggl [ \Biggl ( g \left( x \right) \, q \left( x \right) \Biggr )\, G^{6} + \cdots \Biggr ] + O \left( h^{14} \right) \end{aligned}$$

The \(s-v\) plane for the new obtained four stages symmetric two-step method is shown in Fig. 4.

Fig. 4
figure 4

\(s-v\) plane of the new produced four stages two-step twelfth algebraic order method with vanished phase-lag (phase-fitted)

Full size image

Remark 11

For the case where \(s=v\) (i.e. see the surroundings of the first diagonal of the \(s-v\) plane), the interval of periodicity for this four stages symmetric two-step method is equal to: \(\left( 0, 480 \right) \).

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Liang, M., Simos, T.E. 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, 1187–1211 (2016). https://doi.org/10.1007/s10910-016-0615-x

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  • DOI: https://doi.org/10.1007/s10910-016-0615-x

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