Advertisement

Journal of Mathematical Chemistry

, Volume 56, Issue 8, pp 2454–2484 | Cite as

New Runge–Kutta type symmetric two-step method with optimized characteristics

  • Ke Yan
  • T. E. Simos
Original Paper

Abstract

In this paper and for the first time in the literature, we build a new hybrid symmetric two-step method with the following properties: (1) the new scheme is of symmetric type, (2) the new scheme is of two-step, (3) the new scheme is of five-stages, (4) the new scheme is of twelfth-algebraic order, (5) the new scheme has eliminated the phase-lag and its first, second, third, fourth and fifth derivatives, (6) the new scheme has improved stability characteristics for the general problems, (7) the new scheme is P-stable [with interval of periodicity equal to \(\left( 0, \infty \right) \)] and (8) the new scheme builded based on the following approximations:
  • the first stage is approximation on the point \(x_{n-1}\),

  • the second stage is approximation on the point \(x_{n-1}\),

  • the third stage is approximation on the point \(x_{n-1}\),

  • the fourth stage is approximation on the point \(x_{n}\) and finally,

  • the fifth stage is approximation on the point \(x_{n+1}\),

For the new builded scheme we give a full numerical analysis (local truncation error and stability analysis). The efficiency of the new builded scheme is examined with 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

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

10910_2018_899_MOESM1_ESM.pdf (130 kb)
Supplementary material 1 (pdf 130 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)Google 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)Google Scholar
  3. 3.
    J.M. Franco, M. Palacios, J. Comput. Appl. Math. 30, 1 (1990)Google Scholar
  4. 4.
    J.D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem (John Wiley & Sons Ltd., Baffins Lane, Chichester, West Sussex, England, 1991), pp. 104–107Google Scholar
  5. 5.
    E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)Google 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)Google 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)Google Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175, IX (2005)Google Scholar
  12. 12.
    T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Num. Math. 19, 65–75 (1972)Google 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)Google Scholar
  14. 14.
    R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)Google 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)Google 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)Google 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)Google Scholar
  18. 18.
    Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)Google 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)Google 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)Google 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)Google 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)Google Scholar
  23. 23.
    Ch. Tsitouras, T.E. Simos, On ninth order, explicit Numerov type methods with constant coefficients. Mediterr. J. Math., in pressGoogle 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)Google 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)Google Scholar
  26. 26.
    T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems. Appl. Math. 22(10), 1616–1621 (2009)Google 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)Google 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)Google Scholar
  29. 29.
    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 pages, 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. Volume 2012, Article ID 420387, 17 pages.  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)Google Scholar
  32. 32.
    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)Google Scholar
  33. 33.
    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)Google Scholar
  34. 34.
    Ibraheem 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)Google Scholar
  35. 35.
    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)Google 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)Google Scholar
  37. 37.
    Ibraheem 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)Google Scholar
  38. 38.
    Ibraheem 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
  39. 39.
    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)Google 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)Google 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)Google 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)Google Scholar
  43. 43.
    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)Google 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)Google 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)Google 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 and Applied Analysis Article Number: 910624 Published: 2013Google 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.
    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)Google 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)Google 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)Google Scholar
  51. 51.
    T.E. Simos, Ch. Tsitouras, Evolutionary generation of high order. Explicit two step methods for second order linear IVPs. Math. Methods Appl. Sci. 40, 6276–6284 (2017)Google Scholar
  52. 52.
    T.E. Simos, Ch. Tsitouras, A new family of 7 stages, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 40, 7867–7878 (2017)Google Scholar
  53. 53.
    D.B. Berg, T.E. Simos, C.H. Tsitouras, Trigonometric fitted, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 41, 1845–1854 (2018)Google Scholar
  54. 54.
    T.E. Simos, Ch. Tsitouras, ITh 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)Google Scholar
  56. 56.
    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)Google Scholar
  57. 57.
    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
  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)Google Scholar
  59. 59.
    T. Monovasilis, Z. Kalogiratou, Higinio 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)Google Scholar
  60. 60.
    Thedore 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
  61. 61.
    Z. Kalogiratou, T. Monovasilis, Higinio 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
  62. 62.
    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 72, 1089–1102 (2016)Google Scholar
  63. 63.
    T.H. Monovasilis, Z. Kalogiratou, T.E. Simos, Trigonometrical fitting conditions for two derivative Runge–Kutta methods. Numer. Algorithms in press—online firstGoogle 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 (2003). Article Number: PII S0898-1221(02)00354-1Google 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)Google 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)Google 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)Google 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)Google Scholar
  70. 70.
    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(10), 2191–2213 (2015)Google Scholar
  71. 71.
    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)Google Scholar
  72. 72.
    L. Gr, Ixaru, M. Micu, Topics in Theoretical Physics, Central Institute of Physics, Bucharest (1978)Google Scholar
  73. 73.
    L.Gr 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)Google 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)Google Scholar
  75. 75.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)Google Scholar
  76. 76.
    G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)Google 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)Google 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)Google 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)Google 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)Google 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)Google 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)Google 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)Google 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)Google 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)Google Scholar
  87. 87.
    J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)Google 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)Google Scholar
  89. 89.
    Minjian 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)Google Scholar
  90. 90.
    Xiaopeng 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)Google 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.
    Zhou 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)Google Scholar
  93. 93.
    Fei Hui, Theodore 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.
    Wei 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)Google Scholar
  95. 95.
    L. Zhang, T.E. Simos, An efficient numerical method for the solution of the Schrödinger equation. Adv. Math. Phys. 2016 Article ID 8181927, 20 pages.  https://doi.org/10.1155/2016/8181927
  96. 96.
    Ming DONG, Theodore E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat 31(15), 4999–5012 (2017)Google Scholar
  97. 97.
    Rong-an LIN, Theodore 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
  98. 98.
    Hang 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)Google Scholar
  99. 99.
    Zhiwei 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)Google Scholar
  100. 100.
    Jing Ma, T.E. Simos, An efficient and computational effective method for second order problems. J. Math. Chem. 55, 1649–1668 (2017)Google Scholar
  101. 101.
    Vladislav N. Kovalnogov, Ruslan V. Fedorov, Viktor M. Golovanov, Boris M. Kostishko, T.E. Simos, A four stages numerical pair with optimal phase and stability properties. J. Math. Chem. 56(1), 81–102 (2018)Google Scholar
  102. 102.
    Ke Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem. 56(1), 170–192 (2018)Google Scholar
  103. 103.
    F.A.N.G. Jie, Chenglian Liu, T.E. Simos, A hybric finite difference pair with maximum phase and stability properties. J. Math. Chem. 56(2), 423–448 (2018)Google Scholar
  104. 104.
    Junfeng Yao, T.E. Simos, New finite difference pair with optimized phase and stability properties. J. Math. Chem. 56(2), 449–476 (2018)Google Scholar
  105. 105.
    Jinbin Zheng, Chenglian Liu, T.E. Simos, A new two-step finite difference pair with optimal properties. J. Math. Chem. 56(3), 770–798 (2018)Google Scholar
  106. 106.
    X. Shi, T.E. Simos New five-stages finite difference pair with optimized phase properties. J. Math. Chem., in press—online firstGoogle Scholar
  107. 107.
    C. Liu, T.E. Simos, A five-stages symmetric method with improved phase properties. J. Math. Chem., in press—online firstGoogle Scholar
  108. 108.
    J. Yao, T.E. Simos, New five-stages two-step method with improved characteristics. J. Math. Chem., to appearGoogle Scholar
  109. 109.
    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 pressGoogle Scholar
  110. 110.
    C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)Google Scholar
  111. 111.
    F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)Google Scholar
  112. 112.
    A.R. Leach, Molecular Modelling—Principles and Applications (Pearson, Essex, 2001)Google Scholar
  113. 113.
    P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford University Press, Oxford, 2011)Google Scholar
  114. 114.
    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, in AIP Conference Proceedings, vol. 1863 (2017), p. 560099Google Scholar
  115. 115.
    V.N. Kovalnogov, R.V. Fedorov, M.S. Boyarkin, Method of calculation of a thermolysis and friction of a turbulent disperse flow in nozzles, in AIP Conference Proceedings, vol. 1863 (2017), p. 560015Google Scholar
  116. 116.
    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, in AIP Conference Proceedings, vol. 1863 (2017), p. 560016Google Scholar
  117. 117.
    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, in AIP Conference Proceedings, vol. 1863 (2017), p. 560017Google Scholar
  118. 118.
    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, in AIP Conference Proceedings, vol. 1863 (2017), p. 560018Google Scholar
  119. 119.
    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)Google Scholar
  120. 120.
    V.N. Kovalnogov, R.V. Fedorov, Numerical analysis of the efficiency of film cooling of surface streamlined by supersonic disperse flow, in AIP Conference Proceedings, vol. 1648 (2015), p. 850031Google Scholar
  121. 121.
    V.N. Kovalnogov, R.V. Fedorov, T.V. Karpukhina, E.V. Tsvetova, Numerical analysis of the temperature stratification of the disperse flow, in AIP Conference Proceedings, vol. 1648 (2015), p. 850033Google Scholar
  122. 122.
    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) 49–53 (1998)Google Scholar
  123. 123.
    N. Kovalnogov, and 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. Izvestiya Vysshikh Uchebnykh Zavedenii Aviatsionaya Tekhnika (1) 58–61 (1996)Google Scholar
  124. 124.
    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)Google Scholar
  125. 125.
    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, in AIP Conference Proceedings, vol. 1738 (2016), p. 480004Google Scholar
  126. 126.
    S. Kottwitz, LaTeX Cookbook, Pages 231–236, Packt Publishing Ltd., Birmingham B3 2PB, UK (2015)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Information EngineeringChang’an UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
  3. 3.Group of Modern Computational MethodsUral Federal UniversityEkaterinburgRussian Federation
  4. 4.Department of Automation EngineeringTEI of Sterea HellasPsachnaGreece
  5. 5.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece
  6. 6.AthensGreece

Personalised recommendations