# A new multistep method with optimized characteristics for initial and/or boundary value problems

- 74 Downloads
- 1 Citations

## Abstract

In this paper we introduce, for the first time in the literature, an optimized multistage symmetric two-step method. This method is considered as optimized due to the following reasons: (1) it is of tenth-algebraic order scheme, (2) it has obliterated the phase-lag and its first, second, third and fourth derivatives, (3) it has improved stability characteristics, (4) it is a P-stable method. For the new proposed multistage symmetric two-step method we present a full theoretical investigation consisted of: (1) local truncation error and comparative error analysis, (2) stability analysis and (3) interval of periodicity analysis. The effectiveness of the new builded multistage symmetric two-step method is evaluated on the 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

## References

- 1.A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys.
**6**, 378–391 (1970)CrossRefGoogle Scholar - 2.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, 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*, 1863, 560099 (2017)Google Scholar - 7.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*,**738**, 480004 (2016)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, in
*AIP Conference Proceedings*, 1648, 850033 (2015)Google Scholar - 9.N. Kovalnogov, E. Nadyseva, O. Shakhov and 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.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 - 11.S. Kottwitz, LaTeX Cookbook, pp. 231–236, Packt Publishing Ltd., Birmingham B3 2PB, UK (2015)Google Scholar
- 12.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 - 13.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.
**in press**Google Scholar - 14.V.N. Kovalnogov, R.V. Fedorov, T.E. Simos, New hybrid symmetric two step scheme with optimized characteristics for second order problems. J. Math. Chem. To appearGoogle Scholar
- 15.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.***in press**Google Scholar - 16.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 - 17.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 - 18.J.M. Franco, M. Palacios, J. Comput. Appl. Math.
**30**, 1 (1990)CrossRefGoogle Scholar - 19.J.D. Lambert,
*Numerical Methods for Ordinary Differential Systems: The Initial Value Problem*(Wiley, New York, 1991), pp. 104–107Google Scholar - 20.E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math.
**13**, 154–175 (1969)CrossRefGoogle Scholar - 21.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 - 22.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 - 23.
- 24.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 - 25.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 - 26.T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175(1): IX–IX (2005)Google Scholar
- 27.T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Numer. Math.
**19**, 65–75 (1972)CrossRefGoogle Scholar - 28.R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT
**24**, 225–238 (1984)CrossRefGoogle Scholar - 29.J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl.
**18**, 189–202 (1976)CrossRefGoogle Scholar - 30.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 - 31.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 - 32.Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math.
**158**(1), 75–82 (2003)CrossRefGoogle Scholar - 33.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 - 34.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 - 35.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 - 36.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 - 37.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 CrossRefGoogle Scholar - 38.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 - 39.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 - 40.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 - 41.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 - 42.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 - 43.T.E. Simos, new stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration, in
*Abstract and Applied Analysis*, Volume 2012, Article ID 182536, 15 p. (2012) https://doi.org/10.1155/2012/182536 - 44.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, 17 p. https://doi.org/10.1155/2012/420387 (2012)
- 45.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 - 46.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 - 47.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 - 48.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 - 49.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 - 50.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 - 51.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 - 52.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 - 53.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 - 54.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. J. Math. Chem.
**51**(1), 194–226 (2013)CrossRefGoogle Scholar - 55.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 - 56.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 - 57.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 - 58.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 - 59.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
- 60.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 - 61.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 - 62.Ch. Tsitouras, ITh Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl.
**62**(4), 2101–2111 (2011)CrossRefGoogle Scholar - 63.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 - 64.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)CrossRefGoogle Scholar - 65.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)CrossRefGoogle Scholar - 66.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 - 67.T.E. Simos, Ch. Tsitouras, Fitted modifications of classical Runge–Kutta pairs of orders 5(4). Math. Methods Appl. Sci.
**41**(12), 4549–4559 (2018)CrossRefGoogle Scholar - 68.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) Article Number: 168. https://doi.org/10.1007/s00009-018-1216-7 (2018) - 69.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 - 70.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 - 71.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)CrossRefGoogle Scholar - 72.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 - 73.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 - 74.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 - 75.T.H.E.D.O.R.E.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 - 76.Z. Kalogiratou, Th 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 - 77.H. 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)CrossRefGoogle Scholar - 78.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 - 79.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)CrossRefGoogle Scholar - 80.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 - 81.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 - 82.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 - 83.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 - 84.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 - 85.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 - 86.L.G. Ixaru, M. Micu,
*Topics in Theoretical Physics*(Central Institute of Physics, Bucharest, 1978)Google Scholar - 87.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 - 88.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 - 89.J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math.
**6**, 19–26 (1980)CrossRefGoogle Scholar - 90.G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J.
**100**, 1694–1700 (1990)CrossRefGoogle Scholar - 91.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 - 92.M.M. Chawla, P.S. Rao, An Noumerov-typ method with minimal phase-lag for the integration of second order periodic initial-value problems II: explicit method. J. Comput. Appl. Math.
**15**, 329–337 (1986)CrossRefGoogle Scholar - 93.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 - 94.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 - 95.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 - 96.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 - 97.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 - 98.R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys.
**33**, 795–804 (1960)CrossRefGoogle Scholar - 99.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 - 100.J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math.
**6**, 19–26 (1980)CrossRefGoogle Scholar - 101.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 - 102.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)CrossRefGoogle Scholar - 103.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)CrossRefGoogle Scholar - 104.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 - 105.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 - 106.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 - 107.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 - 108.L. Zhang, T.E. Simos, An efficient numerical method for the solution of the Schrödinger equation,
*Advances in Mathematical Physics***2016**Article ID 8181927, 20 p. https://doi.org/10.1155/2016/8181927 - 109.D. Ming, T.E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat Filomat
**31**(15), 4999–5012 (2017)CrossRefGoogle Scholar - 110.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 - 111.H. Ning, T.E. Simos, A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation. J. Math. Chem.
**53**(6), 1295–1312 (2015)CrossRefGoogle Scholar - 112.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 - 113.J. Ma, T.E. Simos, An efficient and computational effective method for second order problems. J. Math. Chem
**55**, 1649–1668 (2017)CrossRefGoogle Scholar - 114.L. Yang, An efficient and economical high order method for the numerical approximation of the Schrödinger equation. J. Math. Chem.
**55**(9), 1755–1778 (2017)CrossRefGoogle Scholar - 115.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 - 116.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 - 117.J. Fang, C. Liu, T.E. Simos, A hybric finite difference pair with maximum phase and stability properties. J. Math. Chem.
**56**(2), 423–448 (2018)CrossRefGoogle Scholar - 118.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 - 119.J. Zheng, C. Liu, T.E. Simos, A new two-step finite difference pair with optimal properties. J. Math. Chem.
**56**(3), 770–798 (2018)CrossRefGoogle Scholar - 120.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 - 121.C. Liu, T.E. Simos, A five-stages symmetric method with improved phase properties. J. Math. Chem.
**56**(4), 1313–1338 (2018)CrossRefGoogle Scholar - 122.J. Yao, T.E. Simos, New five-stages two-step method with improved characteristics. J. Math. Chem.
**to appear**Google Scholar - 123.K. Yan, T.E. Simos, New Runge–Kutta type symmetric two-step method with optimized characteristics. J. Math. Chem.
**in press-online first**Google Scholar - 124.V.N. Kovalnogov, R.V. Fedorov, A.A. Bondarenko, T.E. Simos, New hybrid two-step method with optimized phase and stability characteristics. J. Math. Chem. (in press)Google Scholar