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High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation. (English) Zbl 1202.81027
Summary: In the present paper we develop a high algebraic order multistep method. The characteristic property of the new proposed method is the requirement of vanishing the phase-lag and its derivatives. The new method is applied for the approximate solution of the radial Schrödinger equation. The efficiency of the new methodology is proved via error analysis and numerical applications.
MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
65L06Multistep, Runge-Kutta, and extrapolation methods
References:
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[59]Simos T.E.: An eighth order method with minimal phase-lag for accuarate computations for the elastic scattering phase-shift problem. Int. J. Mod. Phys. C 7, 825–835 (1996) · doi:10.1142/S0129183196000685
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[69]Simos T.E.: High-algebraic, high-phase-lag methods for accurate computations for the elastic-scattering phase shift problem. Can. J. Phys. 76, 473–493 (1998) · doi:10.1139/cjp-76-6-473
[70]Simos T.E.: High algebraic order methods with minimal phase-lag for accurate solution of the Schrödinger equation. Int. J. Mod. Phys. C 9, 1055–1071 (1998) · Zbl 0948.81529 · doi:10.1142/S0129183198000996
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[74]Simos T.E.: Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrdinger equation. Comput. Chem. 23, 439–446 (1999) · Zbl 05467537 · doi:10.1016/S0097-8485(99)00028-5
[75]Simos T.E.: Explicit eighth order methods for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Commun. 119, 32–44 (1999) · Zbl 1001.65081 · doi:10.1016/S0010-4655(98)00188-X
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[78]Avdelas G., Simos T.E.: Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Phys. Rev. E 62, 1375–1381 (2000) · doi:10.1103/PhysRevE.62.1375
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[87]Avdelas G., Konguetsof A., Simos T.E.: A generator of dissipative methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 148, 59–73 (2002) · Zbl 1196.65119 · doi:10.1016/S0010-4655(02)00468-X
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[90]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
[91]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
[92]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
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[94]Panopoulos G.A., Anastassi Z.A., Simos T.E.: 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)
[95]Simos T.E.: A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46(3), 981–1007 (2009) · Zbl 1183.81060 · doi:10.1007/s10910-009-9553-1
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[97]Simos T.E., Williams P.S.: On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513–554 (1999) · Zbl 0940.65082 · doi:10.1016/S0097-8485(99)00023-6
[98]Anastassi Z.A., Simos T.E.: Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Rep.-Rev. Sect. Phys. Lett. 482, 1–240 (2009)
[99]Vigo-Aguiar J., Simos T.E.: Review of multistep methods for the numerical solution of the radial Schrödinger equation. Int. J. Quantum Chem. 103(3), 278–290 (2005) · doi:10.1002/qua.20495
[100]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
[101]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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