Skip to main content
SpringerLink
Log in

New finite difference pair with optimized phase and stability properties

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

In this paper and for the first time in the literature, a new four-stages symmetric two-step finite difference pair with optimized phase and stability properties is introduced. The new scheme has the following characteristics:

  1. 1.

    is of symmetric type,

  2. 2.

    is of two-step algorithm,

  3. 3.

    is of four-stages,

  4. 4.

    is of tenth-algebraic order,

  5. 5.

    the approximations which produces the new finite difference pair are the following:

    • An approximation developed on the first layer on the point \(x_{n-1}\),

    • An approximation developed on the second layer on the point \(x_{n-1}\),

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

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

  6. 6.

    it has eliminated the phase-lag and its first, second, third and fourth derivatives,

  7. 7.

    it has optimized stability properties,

  8. 8.

    is a P-stable methods since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).

For the new finite difference scheme we describe an error and stability analysis. The examination of the efficiency of the new obtained finite difference pair is based on its application on systems of coupled differential equations arising from the Schrödinger equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. 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 

  2. C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)

    Google Scholar 

  3. F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)

    Google Scholar 

  4. A.R. Leach, Molecular Modelling-Principles and Applications (Pearson, Essex, 2001)

    Google Scholar 

  5. P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford University Press, Oxford, 2011)

    Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. 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)

    Article  Google Scholar 

  9. 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 

  10. 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. Izvestiya Vysshikh Uchebnykh Zavedenii Aviatsionaya Tekhnika 1, 58–61 (1996)

    Google Scholar 

  11. J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. S. Kottwitz, LaTeX Cookbook (Packt Publishing Ltd., Birmingham, 2015), pp. 231–236

    Google Scholar 

  14. T.E. Simos, P.S. Williams, A finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189–205 (1997)

    Article  Google Scholar 

  15. 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 

  16. 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. 1–22 (2017). doi:10.1007/s10910-017-0782-4

  17. K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem. 1–23 (2017). doi:10.1007/s10910-017-0787-z

  18. J. Fang, C. Liu, T.E. Simos, A hybrid finite difference pair with maximum phase and stability properties. J. Math. Chem. 1–26 (2017). doi:10.1007/s10910-017-0793-1

  19. 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 

  20. 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 

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

    Article  Google Scholar 

  22. J.D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem (Wiley, London, 1991), pp. 104–107

    Google Scholar 

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

    Article  Google Scholar 

  24. 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 

  25. 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 

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

  27. 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 

  28. 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)

    Article  CAS  Google Scholar 

  29. T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175(1), IX-IX MAR 1 (2005)

    Article  Google Scholar 

  30. T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Numer. Math. 19, 65–75 (1972)

    Article  Google Scholar 

  31. R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)

    Article  Google Scholar 

  32. 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 

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

    Article  Google Scholar 

  34. 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 

  35. 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 

  36. 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 

  37. 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 

  38. 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 

  39. 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 

  40. 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 

  41. 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 

  42. 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 

  43. 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 

  44. T.E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration. Abstr. Appl. Anal. 2012, 15 (2012). doi:10.1155/2012/182536

    Article  Google Scholar 

  45. 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, 17 (2012). doi:10.1155/2012/420387

    Article  Google Scholar 

  46. 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 

  47. 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)

    Article  CAS  Google Scholar 

  48. 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)

    Article  CAS  Google Scholar 

  49. 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)

    Article  CAS  Google Scholar 

  50. I. Alolyan, T.E. Simos, A family of explicit linear six-step methods with vanished phase-lag and its first derivative. J. Mathe. Chem. 52(8), 2087–2118 (2014)

    Article  CAS  Google Scholar 

  51. 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 

  52. 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 

  53. 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 

  54. 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)

    Article  CAS  Google Scholar 

  55. 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 

  56. 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)

    Article  CAS  Google Scholar 

  57. 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 

  58. 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)

    Article  Google Scholar 

  59. T. 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 

  60. 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 

  61. 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, 910624 (2013)

    Article  Google Scholar 

  62. 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 

  63. 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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  65. 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)

    Article  Google Scholar 

  66. T.E. Simos, Ch. Tsitouras, Evolutionary generation of high order, explicit two step methods for second order linear IVPs. Math. Methods Appl. Scie (2017). doi:10.1002/mma.4454

  67. 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 

  68. 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 

  69. 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)

    Article  Google Scholar 

  70. 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 

  71. 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)

    Article  Google Scholar 

  72. 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)

    Article  Google Scholar 

  73. Theodore 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)

  74. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  76. 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 

  77. 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)00354-1

    Article  Google Scholar 

  78. 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)

    Article  CAS  Google Scholar 

  79. 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 

  80. 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 

  81. 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 

  82. 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)

    Article  CAS  Google Scholar 

  83. 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)

    Article  CAS  Google Scholar 

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

    Google Scholar 

  85. L.G. Ixaru, M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Computer Physics Communications 19, 23–27 (1980)

    Article  Google Scholar 

  86. 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 

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

    Article  Google Scholar 

  88. 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 

  89. 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)

    Article  Google Scholar 

  90. 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 

  91. 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 

  92. A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224–252 (2010)

    Article  CAS  Google Scholar 

  93. 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 

  94. 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 

  95. 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 

  96. 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 

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

    Article  Google Scholar 

  98. K. Mu, T.E. Simos, A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation. J. Math. Chem. 53, 1239–1256 (2015)

    Article  CAS  Google Scholar 

  99. 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)

    Article  CAS  Google Scholar 

  100. 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)

    Article  CAS  Google Scholar 

  101. 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 

  102. 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)

    Article  CAS  Google Scholar 

  103. 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 

  104. 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)

    Article  Google Scholar 

  105. L. Zhang, T.E. Simos, An efficient numerical method for the solution of the Schrödinger equation. Adv. Math. Phys. 2016, 20 (2016). doi:10.1155/2016/8181927

    Google Scholar 

  106. M. Dong, T. E. Simos, A new high algebraic order efficient finite difference method for the solution of the schrödinger equation. Filomat (in press)

  107. 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)

    Article  Google Scholar 

  108. 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)

    Article  CAS  Google Scholar 

  109. 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)

    Article  CAS  Google Scholar 

  110. J. Ma, T.E. Simos, An efficient and computational effective method for second order problems. J. Math. Chem. 55, 1649–1668 (2017)

    Article  CAS  Google Scholar 

Download references

Acknowledgements

Authors whose names appear on the submission have contributed sufficiently to the scientific work and therefore share collective responsibility and accountability for the results.

Author information

Authors and Affiliations

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

    Junfeng Yao

  2. Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641100, People’s Republic of China

    T. E. Simos

  3. Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00, Tripoli, Greece

    T. E. Simos

  4. 10 Konitsis Street, Amfithea, Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

Authors
  1. Junfeng Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. E. Simos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. E. Simos.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Informed consent

Consent to submit has been received explicitly from all co-authors, as well as from the responsible authorities—tacitly or explicitly—at the institute/organization where the work has been carried out, before the work is submitted.

Additional information

T. E. Simos: 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. Working in part time in Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, Greece.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 59 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, J., Simos, T.E. New finite difference pair with optimized phase and stability properties. J Math Chem 56, 449–476 (2018). https://doi.org/10.1007/s10910-017-0803-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-017-0803-3

Keywords

Mathematics Subject Classfication

Search

Navigation