New threestages symmetric sixstep finite difference method with vanished phaselag and its derivatives up to sixth derivative for second order initial and/or boundary value problems with periodical and/or oscillating solutions
 47 Downloads
Abstract
 1.
is a symmetric hybrid (multistages) sixstep method,
 2.
is of threestages,
 3.
is of twelfth algebraic order,
 4.
has vanished the phaselag and
 5.
has vanished the derivatives of the phaselag up to order six.

the construction of the new sixstep pair,

the presentation of the computed local truncation error of the new sixstep pair,
 the comparative error analysis of the new sixstep pair with other sixstep pairs of the same family which are:

the classical sixstep pair of the family (i.e. the sixstep pair with constant coefficients),

the recently proposed sixstep pair of the same family with vanished phaselag and its first derivative,

the recently proposed sixstep pair of the same family with vanished phaselag and its first and second derivatives,

the recently proposed sixstep pair of the same family with vanished phaselag and its first, second and third derivatives,

the recently proposed sixstep pair of the same family with vanished phaselag and its first, second, third and fourth derivatives and finally,

the recently proposed sixstep pair of the same family with vanished phaselag and its first, second, third, fourth and fifth derivatives


the stability and the interval of periodicity analysis for the new obtained sixstep pair and finally

the investigation of the accuracy and computational efficiency of the new developed sixstep pair for the solution of the Schrödinger equation.
Keywords
Schrödinger equation Multistep methods Multistage methods Interval of periodicity Phaselag Phasefitted Derivatives of the phaselagNotes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Supplementary material
References
 1.L.D. Landau, F.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965)Google Scholar
 2.I. Prigogine, S. Rice (eds.), Advances in Chemical Physics, vol. 93: New Methods in Computational Quantum Mechanics (Wiley, New York, 1997)Google Scholar
 3.G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, Toronto, 1950)Google Scholar
 4.T.E. Simos, G. Psihoyios, Special issue: the international conference on computational methods in sciences and engineering 2004—preface. J. Comput. Appl. Math. 191(2), 165 (2006)CrossRefGoogle Scholar
 5.Th. Monovasilis, Z. Kalogiratou, T.E. Simos, Trigonometrical fitting conditions for two derivative Runge–Kutta methods. Numer. Algor. (2017). https://doi.org/10.1007/s1107501704613
 6.Ch. Tsitouras, T.E. Simos, On ninth order, explicit Numerovtype methods with constant coefficients. Mediterr. J. Math. (2018). https://doi.org/10.1007/s0000901810899
 7.D.B. Berg, T.E. Simos, Ch. Tsitouras, Trigonometric fitted, eighthorder explicit Numerovtype methods. Math. Method. Appl. Sci. (2017). https://doi.org/10.1002/mma.4711
 8.T.E. Simos, J. VigoAguiar, A modified phasefitted Runge–Kutta method for the numerical solution of the Schrödinger equation. J. Math. Chem. 30(1), 121–131 (2001)CrossRefGoogle Scholar
 9.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
 10.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
 11.D.F. Papadopoulos, T.E. Simos, A new methodology for the construction of optimized Runge–Kutta–Nyström methods. Int. J. Mod. Phys. C 22(6), 623–634 (2011)CrossRefGoogle Scholar
 12.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
 13.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 (2013)Google Scholar
 14.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7(1), 81–85 (2013)CrossRefGoogle Scholar
 15.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
 16.Z. Kalogiratou, T. Monovasilis, G. Psihoyios, T.E. Simos, Runge–Kutta type methods with special properties for the numerical integration of ordinary differential equations. Phys. Rep. Rev. Sect. Phys. Lett. 536(3), 75–146 (2014)Google Scholar
 17.Z. Kalogiratou, T. Monovasilis, T.E. Simos, A fourth order modified trigonometrically fitted symplectic Runge–Kutta–Nyström method. Comput. Phys. Commun. 185(12), 3151–3155 (2014)CrossRefGoogle Scholar
 18.A.A. Kosti, Z.A. Anastassi, T.E. Simos, An optimized explicit Runge–Kutta method with increased phaselag order for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 47(1), 315–330 (2010)CrossRefGoogle Scholar
 19.Z. Kalogiratou, T.E. Simos, Construction of trigonometrically and exponentially fitted Runge–Kutta–Nyström methods for the numerical solution of the Schrödinger equation and related problems a method of 8th algebraic order. J. Math. Chem. 31(2), 211–232 (2002)CrossRefGoogle Scholar
 20.T.E. Simos, A fourth algebraic order exponentiallyfitted Runge–Kutta method for the numerical solution of the Schrödinger equation. IMA J. Numer. Anal. 21(4), 919–931 (2001)CrossRefGoogle Scholar
 21.T.E. Simos, Exponentiallyfitted Runge–Kutta–Nyström method for the numerical solution of initialvalue problems with oscillating solutions. Appl. Math. Lett. 15(2), 217–225 (2002)CrossRefGoogle Scholar
 22.C. Tsitouras, T.E. Simos, Optimized Runge–Kutta pairs for problems with oscillating solutions. J. Comput. Appl. Math. 147(2), 397–409 (2002)CrossRefGoogle Scholar
 23.Z.A. Anastassi, T.E. Simos, Trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 281–293 (2005)CrossRefGoogle Scholar
 24.Z.A. Anastassi, T.E. Simos, A family of exponentiallyfitted Runge–Kutta methods with exponential order up to three for the numerical solution of the Schrödinger equation. J. Math. Chem. 41(1), 79–100 (2007)CrossRefGoogle Scholar
 25.J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)CrossRefGoogle Scholar
 26.G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
 27.H. Ramos, Z. Kalogiratou, T. Monovasilis, T.E. Simos, An optimized twostep hybrid block method for solving general second order initialvalue problems. Numer. Algorithms 72, 1089–1102 (2016)CrossRefGoogle Scholar
 28.C. Tsitouras, I.T. Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)CrossRefGoogle Scholar
 29.C. Tsitouras, I.T. Famelis, T.E. Simos, Phasefitted RungeŰ–Kutta pairs of orders 8(7). J. Comp. Appl. Math. 321, 226–231 (2017)CrossRefGoogle Scholar
 30.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
 31.C. Tsitouras, T.E. Simos, Evolutionary generation of highorder, explicit, 2step methods for secondorder linear IVPs. Math. Methods Appl. Sci. 40, 6276–6284 (2017)CrossRefGoogle Scholar
 32.C. Tsitouras, T.E. Simos, A new family of 7 stages, eighthorder explicit Numerovtype methods. Math. Methods Appl. Sci. 40, 7867–7878 (2017)CrossRefGoogle Scholar
 33.
 34.G. Avdelas, A. Konguetsof, T.E. Simos, A generator and an optimized generator of highorder hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 1. Development of the basic method. J. Math. Chem. 29(4), 281–291 (2001)CrossRefGoogle Scholar
 35.M.M. Chawla, P.S. Rao, An explicit sixthorder method with phaselag of order eight for \(y^{\prime \prime }=f(t, y)\). J. Comput. Appl. Math. 17, 363–368 (1987)Google Scholar
 36.M.M. Chawla, P.S. Rao, An Noumerovtyp method with minimal phaselag for the integration of second order periodic initialvalue problems II. Explicit method. J. Comput. Appl. Math. 15, 329–337 (1986)CrossRefGoogle Scholar
 37.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
 38.G. Avdelas, A. Konguetsof, T.E. Simos, A generator and an optimized generator of highorder hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 2. Development of the generator; optimization of the generator and numerical results. J. Math. Chem. 29(4), 293–305 (2001)CrossRefGoogle Scholar
 39.T.E. Simos, J. VigoAguiar, Symmetric eighth algebraic order methods with minimal phaselag for the numerical solution of the Schrödinger equation. J. Math. Chem. 31(2), 135–144 (2002)CrossRefGoogle Scholar
 40.A. Konguetsof, T.E. Simos, A generator of hybrid symmetric fourstep methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 93–106 (2003)CrossRefGoogle Scholar
 41.T.E. Simos, I.T. Famelis, C. Tsitouras, Zero dissipative, explicit Numerovtype methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27–40 (2003)CrossRefGoogle Scholar
 42.D.P. Sakas, T.E. Simos, Multiderivative methods of eighth algrebraic order with minimal phaselag for the numerical solution of the radial Schrödinger equation. J. Comput. Appl. Math. 175(1), 161–172 (2005)CrossRefGoogle Scholar
 43.T.E. Simos, Optimizing a class of linear multistep methods for the approximate solution of the radial Schrödinger equation and related problems with respect to phaselag. Cent. Eur. J. Phys. 9(6), 1518–1535 (2011)Google Scholar
 44.D.P. Sakas, T.E. Simos, A family of multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 317–331 (2005)CrossRefGoogle Scholar
 45.H. Van de Vyver, Phasefitted and amplificationfitted twostep hybrid methods for \(y^{\prime \prime }=f(x, y)\). J. Comput. Appl. Math. 209(1), 33–53 (2007)CrossRefGoogle Scholar
 46.H. Van de Vyver, An explicit Numerovtype method for secondorder differential equations with oscillating solutions. Comput. Math. Appl. 53, 1339–1348 (2007)CrossRefGoogle Scholar
 47.T.E. Simos, A new Numerovtype method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46(3), 981–1007 (2009)CrossRefGoogle Scholar
 48.G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A new symmetric eightstep predictor–corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems. Int. J. Mod. Phys. C 22(2), 133–153 (2011)CrossRefGoogle Scholar
 49.G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A symmetric eightstep predictor–corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions. Comput. Phys. Commun. 182(8), 1626–1637 (2011)CrossRefGoogle Scholar
 50.T.E. Simos, Optimizing a hybrid twostep method for the numerical solution of the Schrödinger equation and related problems with respect to phaselag. J. Appl. Math. 2012, Article ID 420387 (2012)Google Scholar
 51.T.E. Simos, A twostep method with vanished phaselag and its first two derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 49(10), 2486–2518 (2011)CrossRefGoogle Scholar
 52.I. Alolyan, T.E. Simos, A family of highorder multistep methods with vanished phaselag and its derivatives for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 62(10), 3756–3774 (2011)CrossRefGoogle Scholar
 53.I. Alolyan, T.E. Simos, A new fourstep hybrid type method with vanished phaselag and its first derivatives for each level for the approximate integration of the Schrödinger equation. J. Math. Chem. 51, 2542–2571 (2013)CrossRefGoogle Scholar
 54.I. Alolyan, T.E. Simos, A Runge–Kutta type fourstep method with vanished phaselag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem. 52, 917–947 (2014)CrossRefGoogle Scholar
 55.I. Alolyan, T.E. Simos, Efficient low computational cost hybrid explicit fourstep method with vanished phaselag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53, 1808–1834 (2015)CrossRefGoogle Scholar
 56.I. Alolyan, T.E. Simos, Family of symmetric linear sixstep methods with vanished phaselag and its derivatives and their application to the radial Schrödinger equation and related problems. J. Math. Chem. 54, 466–502 (2016)CrossRefGoogle Scholar
 57.I. Alolyan, T.E. Simos, A family of two stages tenth algebraic order symmetric sixstep methods with vanished phaselag and its first derivatives for the numerical solution of the radial Schrödinger equation and related problems. J. Math. Chem. 54, 1835–1862 (2016)CrossRefGoogle Scholar
 58.I. Alolyan, T.E. Simos, A new two stages tenth algebraic order symmetric sixstep method with vanished phaselag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems. J. Math. Chem. 55, 105–131 (2017)CrossRefGoogle Scholar
 59.I. Alolyan, T.E. Simos, New two stages high order symmetric sixstep method with vanished phase–lag and its first, second and third derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 55, 503–531 (2017)CrossRefGoogle Scholar
 60.I. Alolyan, Z.A. Anastassi, A new family of symmetric linear fourstep methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput. 218(9), 5370–5382 (2012)Google Scholar
 61.Z.A. Anastassi, T.E. Simos, A parametric symmetric linear fourstep method for the efficient integration of the Schrödinger equation and related oscillatory problems. J. Comput. Appl. Math. 236(16), 3880–3889 (2012)CrossRefGoogle Scholar
 62.G.A. Panopoulos, T.E. Simos, An optimized symmetric 8step semiembedded predictor–corrector method for IVPs with oscillating solutions. Appl. Math. Inf. Sci. 7(1), 73–80 (2013)CrossRefGoogle Scholar
 63.G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A new eightstep symmetric embedded predictor–corrector method (EPCM) for orbital problems and related IVPs with oscillatory solutions. Astron. J. 145(3), Article Number: 75 (2013). https://doi.org/10.1088/00046256/145/3/75
 64.T.E. Simos, New high order multiderivative explicit fourstep methods with vanished phaselag and its derivatives for the approximate solution of the Schrödinger equation. Part I: construction and theoretical analysis. J. Math. Chem. 51(1), 194–226 (2013)CrossRefGoogle Scholar
 65.T.E. Simos, On the explicit fourstep methods with vanished phaselag and its first derivative. Appl. Math. Inf. Sci. 8(2), 447–458 (2014)CrossRefGoogle Scholar
 66.G.A. Panopoulos, T.E. Simos, A new optimized symmetric embedded predictor–corrector method (EPCM) for initialvalue problems with oscillatory solutions. Appl. Math. Inf. Sci. 8(2), 703–713 (2014)CrossRefGoogle Scholar
 67.T.E. Simos, An explicit fourstep method with vanished phaselag and its first and second derivatives. J. Math. Chem. 52(3), 833–855 (2014)CrossRefGoogle Scholar
 68.T.E. Simos, An explicit linear sixstep method with vanished phaselag and its first derivative. J. Math. Chem. 52(7), 1895–1920 (2014)CrossRefGoogle Scholar
 69.T.E. Simos, A new explicit hybrid fourstep method with vanished phaselag and its derivatives. J. Math. Chem. 52(7), 1690–1716 (2014)CrossRefGoogle Scholar
 70.I. Alolyan, T.E. Simos, A family of explicit linear sixstep methods with vanished phaselag and its first derivative. J. Math. Chem. 52(8), 2087–2118 (2014)CrossRefGoogle Scholar
 71.I. Alolyan, T.E. Simos, A hybrid type fourstep method with vanished phaselag 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
 72.T.E. Simos, Multistage symmetric twostep Pstable method with vanished phaselag and its first, second and third derivatives. Appl. Comput. Math. 14(3), 296–315 (2015)Google Scholar
 73.F. Hui, T.E. Simos, Four stages symmetric twostep Pstable method with vanished phaselag and its first, second, third and fourth derivatives. Appl. Comput. Math. 15(2), 220–238 (2016)Google Scholar
 74.I. Alolyan, T.E. Simos, A family of embedded explicit sixstep methods with vanished phaselag and its derivatives for the numerical integration of the Schrödinger equation: development and theoretical analysis. J. Math. Chem. 54(5), 1159–1186 (2016)CrossRefGoogle Scholar
 75.M. Liang, T.E. Simos, A new four stages symmetric twostep method with vanished phaselag and its first derivative for the numerical integration of the Schrödinger equation. J. Math. Chem. 54(5), 1187–1211 (2016)CrossRefGoogle Scholar
 76.I. Alolyan, T.E. Simos, An implicit symmetric linear sixstep methods with vanished phaselag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems. J. Math. Chem. 54(4), 1010–1040 (2016)CrossRefGoogle Scholar
 77.Z. Zhou, T.E. Simos, A new two stage symmetric twostep method with vanished phaselag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 54(2), 442–465 (2016)CrossRefGoogle Scholar
 78.F. Hui, T.E. Simos, A new family of two stage symmetric twostep methods with vanished phaselag and its derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(10), 2191–2213 (2015)CrossRefGoogle Scholar
 79.I. Alolyan, T.E. Simos, A high algebraic order multistage explicit fourstep method with vanished phaselag 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
 80.I. Alolyan, T.E. Simos, A high algebraic order predictor–corrector explicit method with vanished phaselag 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
 81.K. Mu, T.E. Simos, A Runge–Kutta type implicit high algebraic order twostep method with vanished phaselag 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(5), 1239–1256 (2015)CrossRefGoogle Scholar
 82.I. Alolyan, T.E. Simos, A predictor–corrector explicit fourstep method with vanished phaselag 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
 83.I. Alolyan, T.E. Simos, A new eight algebraic order embedded explicit sixstep method with vanished phaselag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 54(8), 1696–1727 (2016)CrossRefGoogle Scholar
 84.T.E. Simos, A new explicit fourstep method with vanished phaselag and its first and second derivatives. J. Math. Chem. 53(1), 402–429 (2015)CrossRefGoogle Scholar
 85.G.A. Panopoulos, T.E. Simos, An eightstep semiembedded 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
 86.A. Konguetsof, A new twostep hybrid method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871–890 (2010)CrossRefGoogle Scholar
 87.K. Tselios, T.E. Simos, Symplectic methods for the numerical solution of the radial Shrödinger equation. J. Math. Chem. 34(1–2), 83–94 (2003)CrossRefGoogle Scholar
 88.K. Tselios, T.E. Simos, Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation. J. Math. Chem. 35(1), 55–63 (2004)CrossRefGoogle Scholar
 89.T. Monovasilis, T.E. Simos, New secondorder exponentially and trigonometrically fitted symplectic integrators for the numerical solution of the timeindependent Schrödinger equation. J. Math. Chem. 42(3), 535–545 (2007)CrossRefGoogle Scholar
 90.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 37(3), 263–270 (2005)CrossRefGoogle Scholar
 91.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 40(3), 257–267 (2006)CrossRefGoogle Scholar
 92.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
 93.T.E. Simos, Closed Newton–Cotes trigonometricallyfitted formulae of highorder for longtime integration of orbital problems. Appl. Math. Lett 22(10), 1616–1621 (2009)CrossRefGoogle Scholar
 94.Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for longtime integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)CrossRefGoogle Scholar
 95.T.E. Simos, High order closed Newton–Cotes trigonometricallyfitted formulae for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 209(1), 137–151 (2009)Google Scholar
 96.T.E. Simos, Closed Newton–Cotes trigonometricallyfitted formulae for the solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 787–801 (2008)Google Scholar
 97.T.E. Simos, Closed Newton–Cotes trigonometricallyfitted formulae of high order for the numerical integration of the Schrödinger equation. J. Math. Chem. 44(2), 483–499 (2008)CrossRefGoogle Scholar
 98.T.E. Simos, Highorder closed Newton–Cotes trigonometricallyfitted formulae for longtime integration of orbital problems. Comput. Phys. Commun. 178(3), 199–207 (2008)CrossRefGoogle Scholar
 99.T.E. Simos, Closed Newton–Cotes trigonometricallyfitted formulae for numerical integration of the Schrödinger equation. Comput. Lett. 3(1), 45–57 (2007)CrossRefGoogle Scholar
 100.T.E. Simos, Closed Newton–Cotes trigonometricallyfitted formulae for longtime integration of orbital problems. RevMexAA 42(2), 167–177 (2006)Google Scholar
 101.T.E. Simos, Closed Newton–Cotes trigonometricallyfitted formulae for longtime integration. Int. J. Mod. Phys. C 14(8), 1061–1074 (2003)CrossRefGoogle Scholar
 102.T.E. Simos, New closed Newton–Cotes type formulae as multilayer symplectic integrators. J. Chem. Phys. 133(10), Article Number: 104108 (2010)Google Scholar
 103.T.E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for longtime integration. Abstr. Appl. Anal. Article Number: 182536 (2012). https://doi.org/10.1155/2012/182536
 104.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
 105.T.E. Simos, Accurately closed Newton–Cotes trigonometricallyfitted formulae for the numerical solution of the Schrödinger equation. Int. J. Mod. Phys. (2013). https://doi.org/10.1142/S0129183113500149
 106.T.E. Simos, New open modified Newton–Cotes type formulae as multilayer symplectic integrators. Appl. Math. Model. 37(4), 1983–1991 (2013)CrossRefGoogle Scholar
 107.G.V. Berghe, M. Van Daele, Exponentially fitted open Newton–Cotes differential methods as multilayer symplectic integrators. J. Chem. Phys. 132, 204107 (2010)CrossRefGoogle Scholar
 108.Z. Kalogiratou, T. Monovasilis, T.E. Simos, A fifthorder symplectic trigonometrically fitted partitioned Runge–Kutta method, in International Conference on Numerical Analysis and Applied Mathematics, SEP 1620, 2007 Corfu, Greece, AIP Conference Proceedings, vol. 936 (2007), pp. 313–317Google Scholar
 109.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems. Comput. Phys. Commun. 177(10), 757–763 (2007)CrossRefGoogle Scholar
 110.T. Monovasilis, T.E. Simos, Symplectic methods for the numerical integration of the Schrödinger equation. Comput. Mater. Sci. 38(3), 526–532 (2007)CrossRefGoogle Scholar
 111.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge–Kutta methods. Phys. Lett. A 372(5), 569–573 (2008)CrossRefGoogle Scholar
 112.Z. Kalogiratou, T. Monovasilis, T.E. Simos, New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation. Comput. Math. Appl. 60(6), 1639–1647 (2010)CrossRefGoogle Scholar
 113.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
 114.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
 115.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Two new phasefitted symplectic partitioned Runge–Kutta methods. Int. J. Mod. Phys. C 22(12), 1343–1355 (2011)CrossRefGoogle Scholar
 116.K. Tselios, T.E. Simos, Optimized fifth order symplectic integrators for orbital problems. Rev. Mex. Astron. Astrofis. 49(1), 11–24 (2013)Google Scholar
 117.Z. Kalogiratou, T. Monovasilis, H. Ramos, T.E. Simos, A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math. 303, 146–155 (2016)CrossRefGoogle Scholar
 118.T. Monovasilis, Z. Kalogiratou, H. Ramos, T.E. Simos, Modified twostep hybrid methods for the numerical integration of oscillatory problems. Math. Methods Appl. Sci. 40(4), 5286–5294 (2017)CrossRefGoogle Scholar
 119.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Symplectic partitioned Runge–Kutta methods with minimal phaselag. Comput. Phys. Commun. 181(7), 1251–1254 (2010)CrossRefGoogle Scholar
 120.L.G. Ixaru, M. Rizea, A Numerovlike 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
 121.A.D. Raptis, A.C. Allison, Exponentialfitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)CrossRefGoogle Scholar
 122.J. VigoAguiar, T.E. Simos, Family of twelve steps exponential fitting symmetric multistep methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 32(3), 257–270 (2002)CrossRefGoogle Scholar
 123.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
 124.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
 125.T.E. Simos, Dissipative trigonometricallyfitted methods for linear secondorder IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004)CrossRefGoogle Scholar
 126.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
 127.G. Avdelas, E. Kefalidis, T.E. Simos, New Pstable eighth algebraic order exponentiallyfitted methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 31(4), 371–404 (2002)CrossRefGoogle Scholar
 128.T.E. Simos, A family of trigonometricallyfitted symmetric methods for the efficient solution of the Schrödinger equation and related problems. J. Math. Chem. 34(1–2), 39–58 (2003)CrossRefGoogle Scholar
 129.T.E. Simos, Exponentially—fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)CrossRefGoogle Scholar
 130.T.E. Simos, A fourstep exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem. 40(3), 305–318 (2006)CrossRefGoogle Scholar
 131.H. Van de Vyver, A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems. Appl. Math. Comput. 189(1), 178–185 (2007)Google Scholar
 132.T.E. Simos, A family of fourstep trigonometricallyfitted methods and its application to the Schrodinger equation. J. Math. Chem. 44(2), 447–466 (2009)CrossRefGoogle Scholar
 133.Z.A. Anastassi, T.E. Simos, A family of twostage twostep methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution. J. Math. Chem. 45(4), 1102–1129 (2009)CrossRefGoogle Scholar
 134.G. Psihoyios, T.E. Simos, Sixth algebraic order trigonometrically fitted predictor–corrector methods for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 37(3), 295–316 (2005)CrossRefGoogle Scholar
 135.G. Psihoyios, T.E. Simos, The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order predictor–corrector methods. J. Math. Chem. 40(3), 269–293 (2006)CrossRefGoogle Scholar
 136.Z. Wang, Pstable linear symmetric multistep methods for periodic initialvalue problems. Comput. Phys. Commun. 171(3), 162–174 (2005)CrossRefGoogle Scholar
 137.T.E. Simos, A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation. J. Math. Chem. 27(4), 343–356 (2000)CrossRefGoogle Scholar
 138.Z.A. Anastassi, T.E. Simos, A family of twostage twostep methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution. J. Math. Chem. 45(4), 1102–1129 (2009)CrossRefGoogle Scholar
 139.C. Tang, W. Wang, H. Yan, Z. Chen, Highorder predictor–corrector of exponential fitting for the Nbody problems. J. Comput. Phys. 214(2), 505–520 (2006)CrossRefGoogle Scholar
 140.G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, Two optimized symmetric eightstep implicit methods for initialvalue problems with oscillating solutions. J. Math. Chem. 46(2), 604–620 (2009)CrossRefGoogle Scholar
 141.S. Stavroyiannis, T.E. Simos, Optimization as a function of the phaselag order of nonlinear explicit twostep Pstable method for linear periodic IVPs. Appl. Numer. Math. 59(10), 2467–2474 (2009)CrossRefGoogle Scholar
 142.S. Stavroyiannis, T.E. Simos, A nonlinear explicit twostep fourth algebraic order method of order infinity for linear periodic initial value problems. Comput. Phys. Commun. 181(8), 1362–1368 (2010)CrossRefGoogle Scholar
 143.Z.A. Anastassi, T.E. Simos, Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Rep. 482, 1–240 (2009)CrossRefGoogle Scholar
 144.R. Vujasin, M. Sencanski, J. RadicPeric, M. Peric, A comparison of various variational approaches for solving the onedimensional vibrational Schrödinger equation. MATCH Commun. Math. Comput. Chem. 63(2), 363–378 (2010)Google Scholar
 145.T.E. Simos, P.S. Williams, On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513–554 (1999)CrossRefGoogle Scholar
 146.L.G. Ixaru, M. Rizea, Comparison of some fourstep methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985)CrossRefGoogle Scholar
 147.J. VigoAguiar, T.E. Simos, Review of multistep methods for the numerical solution of the radial Schrödinger equation. Int. J. Quantum Chem. 103(3), 278–290 (2005)CrossRefGoogle Scholar
 148.W. Zhang, T.E. Simos, A highorder twostep phasefitted method for the numerical solution of the Schrödinger equation. Mediterr. J. Math. 13(6), 5177–5194 (2016)CrossRefGoogle Scholar
 149.D.O.N.G. Ming, T.E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat 31(15), 4999–5012 (2017)CrossRefGoogle Scholar
 150.M.A. Medvedev, T.E. Simos, Two stages sixstep method with eliminated phaselag and its first, second, third and fourth derivatives for the approximation of the Schrödinger equation. J. Math. Chem. 55(4), 961–986 (2017)CrossRefGoogle Scholar
 151.D.B. Berg, T.E. Simos, High order computationally economical sixstep method with vanished phaselag and its derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 55(4), 987–1013 (2017)CrossRefGoogle Scholar
 152.D.B. Berg, T.E. Simos, Three stages symmetric sixstep method with eliminated phaselag and its derivatives for the solution of the Schrödinger equation. J. Math. Chem. 55(5), 1213–1235 (2017)CrossRefGoogle Scholar
 153.D.B. Berg, T.E. Simos, An efficient sixstep method for the solution of the Schrödinger equation. J. Math. Chem. 55(8), 1521–1547 (2017)CrossRefGoogle Scholar
 154.M.A. Medvedev, T.E. Simos, A multistep method with optimal properties for second order differential equations. J. Math. Chem. 56, 1–29 (2018)CrossRefGoogle Scholar
 155.D.B. Berg, T.E. Simos, A new multistep finite difference pair for the Schrödinger equation and related problems. J. Math. Chem. 56(3), 656–686 (2018)CrossRefGoogle Scholar
 156.M.A. Medvedev, T.E. Simos, A new threestages sixstep finite difference pair with optimal phase properties for second order initial and/or boundary value problems with periodical and/or oscillating solutions. J. Math. Chem. (to appear) Google Scholar
 157.S. Kottwitz, LaTeX Cookbook (Packt Publishing Ltd., Birmingham, 2015), pp. 231–236Google Scholar
 158.L.G. Ixaru, M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)Google Scholar
 159.J.R. Dormand, M.E.A. ElMikkawy, P.J. Prince, Families of Runge–Kutta–Nyström formulae. IMA J. Numer. Anal. 7, 235–250 (1987)CrossRefGoogle Scholar
 160.J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar