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Journal of Mathematical Chemistry

, Volume 56, Issue 9, pp 2816–2844 | Cite as

New hybrid symmetric two step scheme with optimized characteristics for second order problems

  • V. N. Kovalnogov
  • R. V. Fedorov
  • T. E. Simos
Original Paper

Abstract

In the present paper, for the first time in the literature, we build a new three-stages symmetric two-step finite difference pair with optimized properties. In more details the new method: (1) is of symmetric type, (2) is of two-step algorithm, (3) is of three-stages—i.e. hybrid or Runge–Kutta type, (4) it is of tenth-algebraic order, (5) it has vanished the phase-lag and its first and second derivatives, (6) it has optimized stability properties for the general problems, (7) it is a P-stable finite difference scheme since it has an interval of periodicity equal to \(\left( 0, \infty \right) \). The new Runge–Kutta type algorithm is builded based on the following approximations:
  • An approximation determined on the first layer on the point \(x_{n-1}\),

  • An approximation determined on the second layer on the point \(x_{n}\) and finally,

  • An approximation determined on the third (final) layer on the point \(x_{n+1}\),

A full theoretical analysis (local truncation error analysis, comparative error analysis and stability and interval of periodicity analysis) is given for the new builded finite difference pair. The effectiveness of the new builded hybrid scheme is evaluated on the numerical solution of systems of coupled differential equations of the Schrödinger type.

Keywords

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

Mathematics Subject Classification

65L05 

Notes

Acknowledgements

The research was funded by a grant of the Russian Foundation for Basic Research (RFBR) for the Project No. 16-38-60114-mol-a-dk.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)CrossRefGoogle Scholar
  2. 2.
    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 53(2), 685–717 (2015)Google Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    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
  6. 6.
    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
  7. 7.
    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
  8. 8.
    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
  9. 9.
    I. Alolyan, T.E. Simos, A high algebraic order predictor-corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 53(7), 1495–1522 (2015)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    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
  12. 12.
    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
  13. 13.
    P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford Univ. Press, Oxford, 2011)Google Scholar
  14. 14.
    D.B. Berg, T.E. Simos, C. Tsitouras, Trigonometric fitted, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 41, 1845–1854 (2018)CrossRefGoogle Scholar
  15. 15.
    R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    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
  18. 18.
    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
  19. 19.
    Z. Chen, C. Liu, T.E. Simos, New three-stages symmetric two step method with improved properties for second order initial/boundary value problems. J. Math. Chem. 56(3), 770–798 (2018)CrossRefGoogle Scholar
  20. 20.
    C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)Google Scholar
  21. 21.
    M. Dong, T.E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat 31(15), 4999–5012 (2017)CrossRefGoogle Scholar
  22. 22.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
  23. 23.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
  24. 24.
    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
  25. 25.
    J. Fang, C. Liu, T.E. Simos, A hybric finite difference pair with maximum phase and stability properties. J. Math. Chem. 56(2), 423–448 (2018)CrossRefGoogle Scholar
  26. 26.
    J.M. Franco, M. Palacios, J. Comput. Appl. Math. 30, 1 (1990)CrossRefGoogle Scholar
  27. 27.
  28. 28.
    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
  29. 29.
    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
  30. 30.
    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
  31. 31.
    L.G. Ixaru, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)Google Scholar
  32. 32.
    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
  33. 33.
    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
  34. 34.
    F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)Google Scholar
  35. 35.
    Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)CrossRefGoogle Scholar
  36. 36.
    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
  37. 37.
    Z. Kalogiratou, T. 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
  38. 38.
    Z. Kalogiratou, T. Monovasilis, H. Ramos, T.E. Simos, A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math. 303, 146–155 (2016)CrossRefGoogle Scholar
  39. 39.
    M. Kenan, 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
  40. 40.
    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 Article Number: PII S0898-1221(02)00354-1 (2003)Google Scholar
  41. 41.
    A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224–252 (2010)CrossRefGoogle Scholar
  42. 42.
    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
  43. 43.
    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
  44. 44.
    S. Kottwitz, LaTeX Cookbook (Packt Publishing Ltd., Birmingham, 2015), pp. 231–236Google Scholar
  45. 45.
    V.N. Kovalnogov, R.V. Fedorov, A.A. Bondarenko, T.E. Simos, New hybrid two-step method with optimized phase and stability characteristics. J. Math. Chem. (in press)Google Scholar
  46. 46.
    N. Kovalnogov, E. Nadyseva, O. Shakhov, V. Kovalnogov, Control of turbulent transfer in the boundary layer through applied periodic effects, Izvestiya Vysshikh Uchebnykh Zavedenii Aviatsionaya Tekhnika(1) (1998) 49–53Google Scholar
  47. 47.
    V.N. Kovalnogov, R.V. Fedorov, T.V. Karpukhina, E.V. Tsvetova, Numerical analysis of the temperature stratification of the disperse flow. AIP Conf. Proc. 1648, 850033 (2015)CrossRefGoogle Scholar
  48. 48.
    V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Modeling and development of cooling technology of turbine engine blades. Int. Rev. Mech. Eng. 9(4), 331–335 (2015)Google Scholar
  49. 49.
    V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Y.A. Khakhalev, A.N. Zolotov, Numerical research of turbulent boundary layer based on the fractal dimension of pressure fluctuations. AIP Conf. Proc. 738, 480004 (2016)CrossRefGoogle Scholar
  50. 50.
    V.N. Kovalnogov, T.E. Simos, I.V. Shevchuk, Perspective of mathematical modeling and research of targeted formation of disperse phase clusters in working media for the next-generation power engineering technologies. AIP Conf. Proc. 1863, 560099 (2017)CrossRefGoogle Scholar
  51. 51.
    V.N. Kovalnogov, R.V. Fedorov, V.M. Golovanov, B.M. Kostishko, T.E. Simos, A four stages numerical pair with optimal phase and stability properties. J. Math. Chem. 56(1), 81–102 (2018)CrossRefGoogle Scholar
  52. 52.
    J.D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem (Wiley, New York, 1991), pp. 104–107Google Scholar
  53. 53.
    J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)CrossRefGoogle Scholar
  54. 54.
    A.R. Leach, Molecular Modelling—Principles and Applications (Pearson, Essex, 2001)Google Scholar
  55. 55.
    M. Liang, T.E. Simos, 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
  56. 56.
    C. Liu, T.E. Simos, A five-stages symmetric method with improved phase properties. J. Math. Chem. 56(4), 1313–1338 (2018)CrossRefGoogle Scholar
  57. 57.
    T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Numer. Math. 19, 65–75 (1972)CrossRefGoogle Scholar
  58. 58.
    J. Ma, T.E. Simos, An efficient and computational effective method for second order problems. J. Math. Chem. 55, 1649–1668 (2017)CrossRefGoogle Scholar
  59. 59.
    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
  60. 60.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7(1), 81–85 (2013)CrossRefGoogle Scholar
  61. 61.
    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
  62. 62.
    T. Monovasilis, Z. Kalogiratou, H. Ramos, T.E. Simos, Modified two-step hybrid methods for the numerical integration of oscillatory problems. Math. Methods Appl. Sci. 40(4), 5286–5294 (2017)CrossRefGoogle Scholar
  63. 63.
    H. Ning, T.E. Simos, A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation. J. Math. Chem. 53(6), 1295–1312 (2015)CrossRefGoogle Scholar
  64. 64.
    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
  65. 65.
    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
  66. 66.
    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
  67. 67.
    G.A. Panopoulos, T.E. Simos, An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math. 290, 1–15 (2015)CrossRefGoogle Scholar
  68. 68.
    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
  69. 69.
    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
  70. 70.
    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, Abstract and Applied Analysis Article Number: 910624 Published (2013)Google Scholar
  71. 71.
    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
  72. 72.
    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
  73. 73.
    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
  74. 74.
    G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astronom. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
  75. 75.
    H. Ramos, Z. Kalogiratou, T. Monovasilis, T.E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms 72, 1089–1102 (2016)CrossRefGoogle Scholar
  76. 76.
    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
  77. 77.
    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
  78. 78.
    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
  79. 79.
    L.I.N. Rong-an, T.E. Simos, A two-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Open Phys. 14, 628–642 (2016)Google Scholar
  80. 80.
    X. Shi, T.E. Simos, New five-stages finite difference pair with optimized phase properties. J. Math. Chem. 56(4), 982–1010 (2018)CrossRefGoogle Scholar
  81. 81.
    T.E. Simos, New Stable Closed Newton–Cotes Trigonometrically Fitted Formulae for Long-Time Integration, Abstract and Applied Analysis, Volume 2012, Article ID 182536 (2012).  https://doi.org/10.1155/2012/182536
  82. 82.
    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, Journal of Applied Mathematics, Volume 2012, Article ID 420387 (2012).  https://doi.org/10.1155/2012/420387
  83. 83.
    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
  84. 84.
    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
  85. 85.
    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
  86. 86.
    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
  87. 87.
    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
  88. 88.
    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
  89. 89.
    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
  90. 90.
    T.E. Simos, New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation, Part I: Construction and theoretical analysis. J. Math. Chem. 51(1), 194–226 (2013)CrossRefGoogle Scholar
  91. 91.
    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
  92. 92.
    T.E. Simos, A new explicit hybrid four-step method with vanished phase-lag and its derivatives. J. Math. Chem. 52(7), 1690–1716 (2014)CrossRefGoogle Scholar
  93. 93.
    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
  94. 94.
    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
  95. 95.
    T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175(1), IX-IX MAR 1 (2005)CrossRefGoogle Scholar
  96. 96.
    T.E. Simos, C. Tsitouras, Evolutionary generation of high order, explicit two step methods for second order linear IVPs. Math. Methods Appl. Sci. 40, 6276–6284 (2017)CrossRefGoogle Scholar
  97. 97.
    T.E. Simos, C. Tsitouras, A new family of 7 stages, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 40, 7867–7878 (2017)CrossRefGoogle Scholar
  98. 98.
    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
  99. 99.
    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
  100. 100.
    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
  101. 101.
    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
  102. 102.
    T.E. Simos, C. Tsitouras, I.T. Famelis, Explicit Numerov type methods with constant coefficients: a review. Appl. Comput. Math. 16(2), 89–113 (2017)Google Scholar
  103. 103.
    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
  104. 104.
    E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)CrossRefGoogle Scholar
  105. 105.
    R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)CrossRefGoogle Scholar
  106. 106.
    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
  107. 107.
    C. Tsitouras, I.T. Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)CrossRefGoogle Scholar
  108. 108.
    C. Tsitouras, I.T. Famelis, T.E. Simos, Phase-fitted Runge–Kutta pairs of orders 8(7). J. Comput. Appl. Math. 321, 226–231 (2017)CrossRefGoogle Scholar
  109. 109.
    C. Tsitouras, T.E. Simos, On ninth order, explicit Numerov type methods with constant coefficients. Medi. J. Math. (2018).  https://doi.org/10.1007/s00009-018-1089-9
  110. 110.
    Z. Wang, T.E. Simos, An economical eighth-order method for the approximation of the solution of the Schrödinger equation. J. Math. Chem. 55, 717–733 (2017)CrossRefGoogle Scholar
  111. 111.
    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
  112. 112.
    K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem. 56(1), 170–192 (2018)CrossRefGoogle Scholar
  113. 113.
    K. Yan, T.E. Simos, New Runge–Kutta type symmetric two-step method with optimized characteristics. J. Math. Chem. (to appear)Google Scholar
  114. 114.
    J. Yao, T.E. Simos, New five-stages two-step method with improved characteristics. J. Math. Chem. 56(6), 1567–1594 (2018)CrossRefGoogle Scholar
  115. 115.
    J. Yao, T.E. Simos, New finite difference pair with optimized phase and stability properties. J. Math. Chem. 56(2), 449–476 (2018)CrossRefGoogle Scholar
  116. 116.
    L. Zhang, T.E. Simos, An Efficient Numerical Method for the Solution of the Schrödinger Equation. Adv. Math. Phys. Article ID 8181927 (2016).  https://doi.org/10.1155/2016/8181927
  117. 117.
    W. Zhang, T.E. Simos, A High-Order Two-Step Phase-Fitted Method for the Numerical Solution of the Schrödinger Equation. Mediterr. J. Math. 13(6), 5177–5194 (2016)CrossRefGoogle Scholar
  118. 118.
    J. Zheng, C. Liu, T.E. Simos, A new two-step finite difference pair with optimal properties. J. Math. Chem. 56(3), 770–798 (2018)CrossRefGoogle Scholar
  119. 119.
    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

Copyright information

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Authors and Affiliations

  • V. N. Kovalnogov
    • 1
  • R. V. Fedorov
    • 1
  • T. E. Simos
    • 2
    • 3
    • 4
    • 5
    • 6
  1. 1.Group of Numerical and Applied Mathematics on Urgent Problems of Energy and Power Engineering, Faculty of Power EngineeringUlyanovsk State Technical UniversityUlyanovskRussia
  2. 2.Dr. T.E. SimosAthensGreece
  3. 3.Department of MathematicsCollege of Sciences, King Saud UniversityRiyadhKingdom of Saudi Arabia
  4. 4.Group of Modern Computational MethodsUral Federal UniversityYekaterinburgRussia
  5. 5.Department of Automation EngineeringTEI of Sterea HellasPsachnaGreece
  6. 6.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece

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