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A multiple stage absolute in phase scheme for chemistry problems

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Abstract

We give attention to the evolvement of newly FD scheme for problems existed in Chemistry.

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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. M.M Chawla, P.S. Rao, High-Accuracy P-stable methods for \(Y^{\prime \prime } = F(T,Y)\). IMA J. Numer. Anal. 5(2), 215–220 (1985) (M.M Chawla, Correction, IMA J. Numer. Anal. 6(2), 252–252(1986))

  7. M.M. Chawla, B. Neta, Families of 2-step 4Th-order P-stable methods for 2Nd-order differential-equations. J. Comput. Appl. Math. 15(2), 213–223 (1986)

    Article  Google Scholar 

  8. M.M Chawla, P.S. Rao, A noumerov-type method with minimal phase-lag for the integration of 2Nd-order periodic initial-value problems. 2. Explicit method. J. Comput. Appl. Math. 15(3), 329–337 (1986)

  9. M.M. Chawla, P.S. Rao, B. Neta, 2-step 4Th-order P-stable methods with phase-lag of order 6 for \(Y^{\prime \prime }=F(T, Y)\). J Comput. Appl. Math. 16(2), 233–236 (1986)

    Article  Google Scholar 

  10. M.M. Chawla, P.S. Rao, An explicit 6Th-order method with phase-lag of order 8 for \(Y^{\prime \prime }=F(T, Y)\). J. Comput. Appl. Math. 17(3), 365–368 (1987)

    Article  Google Scholar 

  11. 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 

  12. 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 

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

    Article  Google Scholar 

  14. 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)

    Article  Google Scholar 

  15. M.M. Chawla, M.A. Al-Zanaidi, A two-stage fourth-order “Almost” P-stable method for oscillatory problems. J. Comput. Appl. Math. 89(1), 115–118 (1998)

    Article  Google Scholar 

  16. 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 

  17. 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 

  18. 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 

  19. L. Yang, T.E. Simos, An efficient and economical high order method for the numerical approximation of the Schrödinger equation. J. Math. Chem. 55(9), 1755–1778 (2017)

    Article  CAS  Google Scholar 

  20. 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)

    Article  CAS  Google Scholar 

  21. K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem. 56(1), 170–192 (2018)

    Article  CAS  Google Scholar 

  22. J.P. Coleman, Numerical-methods for \(Y^{\prime \prime }=F(X, Y)\) via rational-approximations for the cosine. IMA J. Numer. Anal. 9(2), 145–165 (1989)

    Article  Google Scholar 

  23. J.P. Coleman, L. Gr, Ixaru, P-stability and exponential-fitting methods for \(Y^{\prime \prime }=F(X, Y)\). IMA J. Numer. Anal. 16(2), 179–199 (1996)

    Article  Google Scholar 

  24. 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)

    Article  CAS  Google Scholar 

  25. L.Gr. Ixaru, S. Berceanu, Coleman’s method maximally adapted to the Schrödinger-equation. Comput. Phys. Commun. 44(1–2), 11–20 (1987)

  26. L.Gr. Ixaru, M. Rizea, Numerov method maximally adapted to the Schrödinger-equation. J. Comput. Phys. 73(2), 306–324 (1987)

  27. L.Gr. Ixaru, G. Vanden Berghe, H. DeMeyer, M.Van Daele, Four-step exponential-fitted methods for nonlinear physical problems. Comput. Phys. Commun. 100(1–2), 56–70 (1997)

  28. L.Gr. Ixaru, M. Rizea, Four step methods for \(Y^{\prime \prime }=F(X,Y)\). J. Comput. Appl. Math. 79(1), 87–99 (1997)

  29. 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(9), 2591–2616 (2018)

    Article  CAS  Google Scholar 

  30. R. Hao, T.E. Simos, New Runge-Kutta type symmetric two step finite difference pair with improved properties for second order initial and/or boundary value problems. J. Math. Chem. 56(10), 3014–3044 (2018)

    Article  CAS  Google Scholar 

  31. G.-H. Qiu, C. Liu, T.E. Simos, A new multistep method with optimized characteristics for initial and/or boundary value problems. J. Math. Chem. 57(1), 119–148 (2019)

    Article  CAS  Google Scholar 

  32. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Trigonometrical fitting conditions for two derivative Runge-Kutta methods. Numer. Algorithms 79, 787–800 (2018)

    Article  Google Scholar 

  33. G. Wang, T.E. Simos, New multiple stages two-step complete in phase algorithm with improved characteristics for second order initial/boundary value problems. J. Math. Chem. 57(2), 494–515 (2019)

    Article  CAS  Google Scholar 

  34. M.Van Daele, G. Vanden Berghe, H. DeMeyer, L.Gr. Ixaru, Exponential-fitted four-step methods for \(Y^{\prime \prime }=F(X,Y)\). Int. J. Comput. Math. 66(3–4), 299–309 (1998)

  35. L.Gr. Ixaru, G. VandenBerghe, H.DeMeyer, Exponentially fitted variable two-step BDF algorithm for first order odes. Comput. Phys. Commun. 150(2), 116–128 (2003)

  36. M. Rizea, Exponential fitting method for the time-dependent Schrödinger equation. J. Math. Chem. 48(1), 55–65 (2010)

    Article  CAS  Google Scholar 

  37. L.Gr. Ixaru, M. Rizea, G. VandenBerghe, H.DeMeyer, Weights of the exponential fitting multistep algorithms for first-order odes. J. Comput. Appl. Math. 132(1), 83–93 (2001)

  38. M.Van Daele, G. Vanden Berghe, H.DeMeyer, L.Gr. Ixaru, Exponential-Fitted Four-Step Methods for \(Y^{\prime \prime }=F(X,Y)\). Int. J. Comput. Math. 66(3–4), 299–309 (1998)

  39. L.Gr. Ixaru, B. Paternoster, A Conditionally P-Stable Fourth-Order Exponential-Fitting Method for \(Y^{\prime \prime } = F(X, Y)\). J. Comput. Appl. Math. 106(1), 87–98 (1999)

  40. 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)

    Article  Google Scholar 

  41. 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 

  42. 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 

  43. 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 

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

    Google Scholar 

  45. M. Xu, T.E. Simos, A multistage two-step fraught in phase scheme for problems in mathematical chemistry. J. Math. Chem. 57(7), 1710–1731 (2019)

    Article  CAS  Google Scholar 

  46. 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 

  47. M.A. Medvedev, T.E. Simos, A three-stages multistep teeming in phase algorithm for computational problems in chemistry. J. Math. Chem. 57(6), 1598–1617 (2019)

    Article  CAS  Google Scholar 

  48. J. Lv, T.E. Simos, A Runge–Kutta type crowded in phase algorithm for quantum chemistry problems. J. Math. Chem. (in press)

  49. 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 

  50. 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 

  51. J.M. Franco, M. Palacios, High-order p-stable multistep methods. J. Comput. Appl. Math. 30, 1 (1990)

    Article  Google Scholar 

  52. J.D. Lambert, Numerical Methods for Ordinary Differential Systems, the Initial Value Problem (Wiley, Hoboken, 1991), pp. 104–107

    Google Scholar 

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

    Article  Google Scholar 

  54. A.D. Raptis, J.R. Cash, A variable step method for the numerical-integration of the one-dimensional Schrödinger-equation. Comput. Phys. Commun. 36(2), 113–119 (1985)

    Article  CAS  Google Scholar 

  55. A.D. Raptis, J.R. Cash, Exponential and bessel fitting methods for the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 44(1–2), 95–103 (1987)

    Article  Google Scholar 

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

  57. J.P. Coleman, A.S. Booth, Analysis of a family of chebyshev methods for \(Y^{\prime \prime } = F(X, Y)\). J. Comput. Appl. Math. 44(1), 95–114 (1992)

    Article  Google Scholar 

  58. J.P. Coleman, S.C. Duxbury, Mixed collocation methods for \(Y^{\prime \prime } = F(X, Y)\). J. Comput. Appl. Math. 126(1–2), 47–75 (2000)

    Article  Google Scholar 

  59. C. Liu, C.-W. Hsu, Ch. Tsitouras, T.E. Simos, Hybrid numerov-type methods with coefficients trained to perform better on classical orbits. Bull. Malays. Math. Sci. Soc. 42(5), 2119–2134 (2019). https://doi.org/10.1007/s40840-019-00775-z

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  62. 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 

  63. 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 

  64. C. Liu, C.-W. Hsu, T.E. Simos, Ch. Tsitouras, Phase-fitted, six-step methods for solving \(x^{\prime \prime }=f(t, x)\). Math. Methods Appl. Sci. 42(11), 3942–3949 (2019)

    Article  Google Scholar 

  65. C. Lin, J.J. Chen, T.E. Simos, Ch. Tsitouras, Evolutionary derivation of sixth-order P-stable SDIRKN methods for the solution of PDEs with the method of lines. Mediterr. J. Math. 16(3), 69 (2019). https://doi.org/10.1007/s00009-019-1336-8

    Article  Google Scholar 

  66. A.D. Raptis, On the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 24(1), 1–4 (1981)

    Article  Google Scholar 

  67. Z. Kalogiratou, T. Monovasilis, T.E. Simos, New fifth order two-derivative Runge-Kutta methods with constant and frequency dependent coefficients. Math. Methods Appl. Sci. 42(6), 1955–1966 (2019)

    Article  Google Scholar 

  68. M.A. Medvedev, T.E. Simos, C. Tsitouras, Hybrid, phase-fitted, four-step methods of seventh order for solving x”(t) = f (t, x). Math. Methods Appl. Sci. (in press)

  69. A.D. Raptis, 2-step methods for the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 28(4), 373–378 (1982)

    Google Scholar 

  70. A.D. Raptis, Exponential-fitting methods for the numerical-integration of the 4Th-order differential-equation \(Yiv+F.Y=G\). Computing 24(2–3), 241–250 (1980)

  71. 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 

  72. 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 

  73. Z. Wang, Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Duffing equation. Comput. Phys. Commun. 174(2), 109–118 (2006)

    Article  CAS  Google Scholar 

  74. 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 

  75. 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 

  76. 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 

  77. 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 

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

    Article  Google Scholar 

  79. 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, 420387 (2012). https://doi.org/10.1155/2012/420387

    Article  Google Scholar 

  80. 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 

  81. 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 

  82. 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 

  83. 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)

    Article  CAS  Google Scholar 

  84. 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 

  85. 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 

  86. 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 

  87. 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 

  88. 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 

  89. 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)

  90. 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 

  91. 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 

  92. 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 

  93. 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 

  94. 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. Article Number: 910624 Published: (2013)

  95. 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 

  96. 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 

  97. 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 

  98. 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 

  99. M.A. Medvedev, T.E. Simos, C. Tsitouras, Trigonometric-fitted hybrid four-step methods of sixth order for solving \(y^{\prime \prime }(x)=f(x, y)\). Math. Methods Appl. Sci 42(2), 710–716 (2019)

    Article  Google Scholar 

  100. M.A. Medvedev, T.E. Simos, C. Tsitouras, Hybrid, phase-fitted, four-step methods of seventh order for solving x”(t) = f (t, x). Math. Methods Appl. Sci. 42(6), 2025–2032 (2019)

    Article  Google Scholar 

  101. Maxim A. Medvedev, T. E. Simos, Ch. Tsitouras, Hybrid, phase-fitted, four-step methods of seventh order for solving x”(t) = f (t, x). Math. Methods Appl. Sci. (to appear)

  102. Ch. Tsitouras, T.E. Simos, On ninth order, explicit Numerov type methods with constant coefficients. Mediterr. J Math. 15(2), 46 (2018). https://doi.org/10.1007/s00009-018-1089-9

    Article  Google Scholar 

  103. 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)

    Article  Google Scholar 

  104. 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)

    Article  Google Scholar 

  105. D.B. Berg, T.E. Simos, C. Tsitouras, Trigonometric fitted, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 41, 1845–1854 (2018)

    Article  Google Scholar 

  106. T.E. Simos, C. Tsitouras, Fitted modifications of classical Runge-Kutta pairs of orders 5 (4). Math. Methods Appl. Sci. 41(12), 4549–4559 (2018)

    Article  Google Scholar 

  107. Ch. Tsitouras, T.E. Simos, Trigonometric fitted explicit Numerov type method with vanishing phase-lag and its first and second derivatives. Mediterr. J. Math. 15(4), 168 (2018). https://doi.org/10.1007/s00009-018-1216-7

    Article  Google Scholar 

  108. M.A. Medvedev, T.E. Simos, C. Tsitouras, Fitted modifications of Runge-Kutta pairs of orders 6 (5). Math. Methods Appl. Sci. 41(16), 6184–6194 (2018)

    Article  Google Scholar 

  109. M.A. Medvedev, T.E. Simos, C. Tsitouras, Explicit, two stage, sixth order, hybrid four-step methods for solving \(y^{\prime \prime }(x)=f(x, y)\). Math. Methods Appl. Sci. 41(16), 6997–7006 (2018)

    Article  Google Scholar 

  110. 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 

  111. T.E. Simos, C. Tsitouras, High phase-lag order, four-step methods for solving \(y^{\prime \prime }=f(x, y)\). Appl. Comput. Math. 17(3), 307–316 (2018)

    Google Scholar 

  112. 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 

  113. 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 

  114. 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 

  115. 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 

  116. 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 

  117. 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 

  118. 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 

  119. 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 

  120. 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 

  121. A. Konguetsof, T.E. Simos, An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems. Comput. Math. Appl. 45(1–3), 547–554 (2003)

    Article  Google Scholar 

  122. 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 

  123. 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 

  124. 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 

  125. 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 

  126. 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 

  127. 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 

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

    Google Scholar 

  129. L.G. Ixaru, M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19, 23–27 (1980)

    Article  Google Scholar 

  130. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  133. 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 

  134. 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 

  135. 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 

  136. 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 

  137. 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 

  138. 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 

  139. 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 

  140. 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 

  141. 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 

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

    Article  Google Scholar 

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Authors and Affiliations

  1. Unmanned System Research Institute, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China

    Xunying Zhang

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

    T. E. Simos

  3. Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Dongtong Road 705, Neijiang, 641100, People’s Republic of China

    T. E. Simos

  4. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece

    T. E. Simos

  5. 10 Konitsis Street, Amphithea - Paleon Phaleron, 175 64, Athens, Greece

    T. E. Simos

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Zhang, X., Simos, T.E. A multiple stage absolute in phase scheme for chemistry problems. J Math Chem 57, 2049–2074 (2019). https://doi.org/10.1007/s10910-019-01054-9

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