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Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. (English) Zbl 1198.81092
Summary: We present a new method for the numerical solution of the time-independent Schrödinger equation for one spatial dimension and related problems. A technique, based on the phase-lag and its derivatives, is used, in order to calculate the parameters of the new Numerov-type algorithm. We study the relation of the local truncation error with the energy of the model of the radial Schrödinger equation and via this investigation we examine how accurate is the new method compared with other well known numerical methods in the literature. We present also the stability analysis of the new method and the relation of the interval of periodicity with the frequency of the test problem and the frequency of the new developed method. We illustrate the accuracy and computational efficiency of the new developed method via numerical examples.
MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
81T80Simulation and numerical modelling (quantum field theory)
65L06Multistep, Runge-Kutta, and extrapolation methods
Software:
pythNon; SCHOL; VFGEN
References:
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[115]Sakas D.P., Simos T.E.: 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) · Zbl 1063.65067 · doi:10.1016/j.cam.2004.06.013
[116]Tselios K., Simos T.E.: Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. J. Comput. Appl. Math. 175(1), 173–181 (2005) · Zbl 1063.65113 · doi:10.1016/j.cam.2004.06.012
[117]Kalogiratou Z., Simos T.E.: Newton-Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003) · Zbl 1041.65104 · doi:10.1016/S0377-0427(03)00479-5
[118]Kalogiratou Z., Monovasilis T., Simos T.E.: Symplectic integrators for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 83–92 (2003) · Zbl 1027.65171 · doi:10.1016/S0377-0427(03)00478-3
[119]Konguetsof A., Simos T.E.: 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) · Zbl 1027.65094 · doi:10.1016/S0377-0427(03)00469-2
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[121]Tsitouras Ch., Simos T.E.: Optimized Runge-Kutta pairs for problems with oscillating solutions. J. Comput. Appl. Math. 147(2), 397–409 (2002) · Zbl 1013.65073 · doi:10.1016/S0377-0427(02)00475-2
[122]Simos T.E.: An exponentially fitted eighth-order method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 108(1–2), 177–194 (1999) · Zbl 0956.65063 · doi:10.1016/S0377-0427(99)00109-0
[123]Simos T.E.: An accurate finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 91(1), 47–61 (1998) · Zbl 0934.65084 · doi:10.1016/S0377-0427(98)00014-4
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[125]Simos T.E., Williams P.S.: A finite-difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79(2), 189–205 (1997) · Zbl 0877.65054 · doi:10.1016/S0377-0427(96)00156-2
[126]Avdelas G., Simos T.E.: A generator of high-order embedded P-stable methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 72(2), 345–358 (1996) · Zbl 0863.65042 · doi:10.1016/0377-0427(96)00005-2
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[128]Simos T.E.: A family of 4-step exponentially fitted pedictor-corrector methods for the numerical-integration of The Schrödinger-equation. J. Comput. Appl. Math. 58(3), 337–344 (1995) · Zbl 0833.65082 · doi:10.1016/0377-0427(93)E0274-P
[129]Simos T.E.: An explicit 4-step phase-fitted method for the numerical-integration of 2nd-order initial-value problems. J. Comput. Appl. Math. 55(2), 125–133 (1994) · Zbl 0823.65067 · doi:10.1016/0377-0427(94)90015-9
[130]Simos T.E., Dimas E., Sideridis A.B.: A Runge-Kutta-Nyström method for the numerical-integration of special 2nd-order periodic initial-value problems. J. Comput. Appl. Math. 51(3), 317–326 (1994) · Zbl 0872.65066 · doi:10.1016/0377-0427(92)00114-O
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[132]Simos T.E., Raptis A.D.: A 4th-order Bessel fitting method for the numerical-solution of the SchrÖdinger-equation. J. Comput. Appl. Math. 43(3), 313–322 (1992) · Zbl 0763.65066 · doi:10.1016/0377-0427(92)90017-R
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[135]Papageorgiou C.D., Raptis A.D., Simos T.E.: A method for computing phase-shifts for scattering. J. Comput. Appl. Math. 29(1), 61–67 (1990) · Zbl 0685.65080 · doi:10.1016/0377-0427(90)90195-6
[136]Raptis A.D.: Two-step methods for the numerical solution of the Schrödinger equation. Computing 28, 373–378 (1982) · Zbl 0473.65060 · doi:10.1007/BF02279820
[137]Simos T.E.: Two-step almost P-stable complete in phase methods for the numerical integration of second order periodic initial-value problems. Int. J. Comput. Math. 46, 77–85 (1992) · Zbl 0812.65067 · doi:10.1080/00207169208804140
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[140]Simos T.E.: A modified Runge-Kutta method for the numerical solution of ODE’s with oscillation solutions. Appl. Math. Lett. 9(6), 61–66 (1996) · Zbl 0864.65052 · doi:10.1016/0893-9659(96)00095-X
[141]SIMOS T.E.: A High-order predictor–corrector method for periodic IVPs. Appl. Math. Lett. 6(5), 9–12 (1993) · Zbl 0782.65094 · doi:10.1016/0893-9659(93)90090-A
[142]SIMOS T.E.: A new variable-step method for the numerical-integration of special 2Nd-order initial-value problems and their application to the one-dimensional SchrÖdinger-equation. Appl. Math. Lett. 6(3), 67–73 (1993) · Zbl 0772.65048 · doi:10.1016/0893-9659(93)90037-N
[143]Simos T.E.: Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems. Appl. Math. Lett. 22(10), 1616–1621 (2009) · Zbl 1171.65449 · doi:10.1016/j.aml.2009.04.008
[144]Papadopoulos D.F., Anastassi Z.A., Simos T.E.: A phase-fitted Runge-Kutta-Nystrom method for the numerical solution of initial value problems with oscillating solutions. Comput. Phys. Commun. 180(10), 1839–1846 (2009) · Zbl 1197.65086 · doi:10.1016/j.cpc.2009.05.014
[145]Simos T.E., Zdetsis A.D., Psihoyios G., Anastassi Z.A.: Special issue on mathematical chemistry based on papers presented within ICCMSE 2005 preface. J. Math. Chem. 46(3), 727–728 (2009) · doi:10.1007/s10910-009-9563-z
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[148]T.E. Simos, Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation, Acta. Appl. Math., (in press)
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[152]Simos T.E.: A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. IMA J. Numer. Anal. 21(4), 919–931 (2001) · Zbl 0990.65079 · doi:10.1093/imanum/21.4.919
[153]Simos T.E.: Some new 4-step exponential-fitting methods for the numerical-solution of the radial SchrÖdinger-equation. IMA J. Numer. Anal. 11(3), 347–356 (1991) · Zbl 0728.65067 · doi:10.1093/imanum/11.3.347
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[155]Simos T.E., Famelis I.T., Tsitouras Ch.: Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27–40 (2003) · Zbl 1031.65080 · doi:10.1023/A:1026167824656
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[158]Simos T.E., Psihoyios G.: Special issue: The international conference on computational methods in sciences and engineering 2004–Preface. J. Comput. Appl. Math. 191(2), 165–165 (2006) · doi:10.1016/j.cam.2005.09.005
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[160]Simos T.E., Vigo-Aguiar J.: Special issue–selected papers from the conference on computational and mathematical methods for science and engineering (CMMSE-2002)–Alicante University, Spain, 20–25 September 2002–Preface. J. Comput. Appl. Math. 158(1), IX–IX (2003) · Zbl 1028.00531 · doi:10.1016/S0377-0427(03)00459-X
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