Advertisement

Journal of Mathematical Chemistry

, Volume 56, Issue 8, pp 2302–2340 | Cite as

New hybrid two-step method with optimized phase and stability characteristics

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

Abstract

In this paper and for the first time in the literature, we develop a new Runge–Kutta type symmetric two-step finite difference pair with the following characteristics:
  • the new algorithm is of symmetric type,

  • the new algorithm is of two-step,

  • the new algorithm is of five-stages,

  • the new algorithm is of twelfth-algebraic order,

  • the new algorithm is based on the following approximations:
    1. 1.

      the first layer on the point \(x_{n-1}\),

       
    2. 2.

      the second layer on the point \(x_{n-1}\),

       
    3. 3.

      the third layer on the point \(x_{n-1}\),

       
    4. 4.

      the fourth layer on the point \(x_{n}\) and finally,

       
    5. 5.

      the fifth (final) layer on the point \(x_{n+1}\),

       
  • the new algorithm has vanished the phase-lag and its first, second, third and fourth derivatives,

  • the new algorithm has improved stability characteristics for the general problems,

  • the new algorithm is of P-stable type since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).

For the new developed algorithm we present a detailed numerical analysis (local truncation error and stability analysis). The effectiveness of the new developed algorithm is evaluated with the approximate solution of coupled differential equations arising from 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

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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.
    J.M. Franco, M. Palacios, J. Comput. Appl. Math. 30, 1 (1990)CrossRefGoogle Scholar
  4. 4.
    J.D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem (Wiley, New York, 1991), pp. 104–107Google Scholar
  5. 5.
    E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    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
  8. 8.
  9. 9.
    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
  10. 10.
    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
  11. 11.
    T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175(1), 1–9 (2005)CrossRefGoogle Scholar
  12. 12.
    T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Numer. Math. 19, 65–75 (1972)CrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)CrossRefGoogle Scholar
  15. 15.
    J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    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
  18. 18.
    Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    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
  21. 21.
    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
  22. 22.
    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
  23. 23.
    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
  24. 24.
    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
  25. 25.
    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
  26. 26.
    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
  27. 27.
    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
  28. 28.
    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
  29. 29.
    T. E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration. Abstract Appl. Anal. 2012, Article ID 182536 (2012).  https://doi.org/10.1155/2012/182536
  30. 30.
    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, Article ID 420387.  https://doi.org/10.1155/2012/420387, (2012)
  31. 31.
    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
  32. 32.
    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
  33. 33.
    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
  34. 34.
    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
  35. 35.
    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
  36. 36.
    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
  37. 37.
    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
  38. 38.
    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)CrossRefGoogle Scholar
  39. 39.
    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
  40. 40.
    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
  41. 41.
    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
  42. 42.
    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
  43. 43.
    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
  44. 44.
    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
  45. 45.
    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
  46. 46.
    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 Appl. Anal. Article Number: 910624 Published (2013)Google Scholar
  47. 47.
    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
  48. 48.
    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
  49. 49.
    Ch. Tsitouras, ITh Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)CrossRefGoogle Scholar
  50. 50.
    Ch. Tsitouras, ITh Famelis, T.E. Simos, Phase-fitted Runge–Kutta pairs of orders 8(7). J. Comput. Appl. Math. 321, 226–231 (2017)CrossRefGoogle Scholar
  51. 51.
    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
  52. 52.
    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
  53. 53.
    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
  54. 54.
    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
  55. 55.
    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
  56. 56.
    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
  57. 57.
    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
  58. 58.
    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
  59. 59.
    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
  60. 60.
    T. 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
  61. 61.
    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
  62. 62.
    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
  63. 63.
    T. Monovasilis, Z. Kalogiratou , T.E. Simos, Trigonometrical fitting conditions for two derivative Runge–Kutta methods. Numer. Algorithms (in press - online first)Google Scholar
  64. 64.
    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
  65. 65.
    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
  66. 66.
    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
  67. 67.
    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
  68. 68.
    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
  69. 69.
    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
  70. 70.
    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
  71. 71.
    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
  72. 72.
    L.G. Ixaru, M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)Google Scholar
  73. 73.
    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
  74. 74.
    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
  75. 75.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
  76. 76.
    G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
  77. 77.
    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
  78. 78.
    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)CrossRefGoogle Scholar
  79. 79.
    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
  80. 80.
    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
  81. 81.
    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
  82. 82.
    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
  83. 83.
    A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)CrossRefGoogle Scholar
  84. 84.
    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
  85. 85.
    R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)CrossRefGoogle Scholar
  86. 86.
    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
  87. 87.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
  88. 88.
    Mu 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
  89. 89.
    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
  90. 90.
    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
  91. 91.
    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
  92. 92.
    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
  93. 93.
    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
  94. 94.
    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
  95. 95.
    L. Zhang, T.E. Simos, An efficient numerical method for the solution of the Schrödinger equation. Adv. Math. Phys. 8181927, 20 (2016).  https://doi.org/10.1155/2016/8181927 CrossRefGoogle Scholar
  96. 96.
    D. Ming, 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
  97. 97.
    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. 14, 628–642 (2016)CrossRefGoogle Scholar
  98. 98.
    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, 1295–1312 (2015)CrossRefGoogle Scholar
  99. 99.
    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
  100. 100.
    J. Ma, T.E. Simos, An efficient and computational effective method for second order problems. J. Math. Chem. 55, 1649–1668 (2017)CrossRefGoogle Scholar
  101. 101.
    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
  102. 102.
    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
  103. 103.
    F. Jie, 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
  104. 104.
    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
  105. 105.
    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
  106. 106.
    C. Liu, T.E. Simos, A five-stages symmetric method with improved phase properties. J. Math. Chem. 56(4), 1313–1338 (2018)CrossRefGoogle Scholar
  107. 107.
    J. Yao, T.E. Simos, New five-stages two-step method with improved characteristics. J. Math. Chem. (2018).  https://doi.org/10.1007/s10910-018-0874-9
  108. 108.
    C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)Google Scholar
  109. 109.
    F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)Google Scholar
  110. 110.
    A.R. Leach, Molecular Modelling—Principles and Applications (Pearson, Essex, 2001)Google Scholar
  111. 111.
    P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford University Press, Oxford, 2011)Google Scholar
  112. 112.
    T.E. Simos, V.N. Kovalnogov, 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
  113. 113.
    V.N. Kovalnogov, R.V. Fedorov, M.S. Boyarkin, Method of calculation of a thermolysis and friction of a turbulent disperse flow in nozzles. AIP Conf. Proc. 1863, 560015 (2017)CrossRefGoogle Scholar
  114. 114.
    V.N. Kovalnogov, R.V. Fedorov, L.V. Khakhaleva, A.V. Chukalin, A.A. Bondarenko, E.N. Kovrizhnykh, The mechanism and theoretical basis of the management of intensity of the heat transfer control through periodic influences on the turbulent boundary layer. AIP Conf. Proc. 1863, 560016 (2017)CrossRefGoogle Scholar
  115. 115.
    V.N. Kovalnogov, R.V. Fedorov, L.V. Khakhaleva, A.N. Zolotov, The modeling of influence of the external turbulence over the heat transfer towards the surface of turbomachinery blades. AIP Conf. Proc. 1863, 560017 (2017)CrossRefGoogle Scholar
  116. 116.
    V.N. Kovalnogov, R.V. Fedorov, Y.A. Khakhalev, L.V. Khakhaleva, A.V. Chukalin, Application of the results of experimental and numerical turbulent flow researches based on pressure pulsations analysis. AIP Conf. Proc. 1863, 560018 (2017)CrossRefGoogle Scholar
  117. 117.
    V.N. Kovalnogov, R.V. Fedorov, L.V. Khakhaleva, D.A. Generalov, A.V. Chukalin, Development and investigation of the technologies involving thermal protection of surfaces immersed in disperse working medium flow. Int. J. Energy Clean Environ. 17(2–4), 223–239 (2016)CrossRefGoogle Scholar
  118. 118.
    V.N. Kovalnogov, R.V. Fedorov, Numerical analysis of the efficiency of film cooling of surface streamlined by supersonic disperse flow. AIP Conf. Proc. 1648, 850031 (2015)CrossRefGoogle Scholar
  119. 119.
    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
  120. 120.
    N. Kovalnogov, E. Nadyseva, O. Shakhov, V. Kovalnogov, Control of turbulent transfer in the boundary layer through applied periodic effects. Izvest. Vysshikh Uchebnykh Zaved. Aviat. Tekh. (1), 49–53 (1998)Google Scholar
  121. 121.
    N. Kovalnogov, V. Kovalnogov, Characteristics of numerical integration and conditions of solution stability in the system of differential equations of boundary layer, subjected to the intense influence. Izvest. Vysshikh Uchebnykh Zaved. Aviat. Tekh. (1), 58–61 (1996)Google Scholar
  122. 122.
    M.S. Boyarkin, V.N. Kovalnogov, T.V. Karpukhina, R.V. Fedorov, Development and research of the technology of enriching low-grade solid fuels with recirculating flue gases for boiler plants. Int. J. Energy Clean Environ. 17(2–4), 145–163 (2016)CrossRefGoogle Scholar
  123. 123.
    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. 1738, 480004 (2016)CrossRefGoogle Scholar
  124. 124.
    S. Kottwitz, LaTeX Cookbook (Packt Publishing Ltd., Birmingham, 2015), pp. 231–236Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • V. N. Kovalnogov
    • 1
  • R. V. Fedorov
    • 1
  • A. A. Bondarenko
    • 2
  • T. E. Simos
    • 3
    • 4
    • 5
    • 6
    • 7
  1. 1.Group of Numerical and Applied Mathematics on Urgent Problems of Energy and Power Engineering, Faculty of Power EngineeringUlyanovsk State Technical UniversityUlyanovskRussian Federation
  2. 2.Ulyanovsk Institute of Civil Aviation named after Chief Marshal of Aviation BP Bugaev State Technical UniversityUlyanovskRussian Federation
  3. 3.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
  4. 4.Group of Modern Computational MethodsUral Federal UniversityEkaterinburgRussian Federation
  5. 5.Department of Automation EngineeringTEI of Sterea HellasPsachnaGreece
  6. 6.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece
  7. 7.AthensGreece

Personalised recommendations