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A family of ten-step methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation. (English) Zbl 1252.81133
Summary: A family ten-step methods of high algebraic order is obtained. The new developed methods have vanished phase-lag (the first one) and phase-lag and its first derivative (the second one). We apply the new developed methods to the resonance problem of the radial Schrödinger equation. The efficiency of the new proposed methods is shown via an error analysis and numerical applications.
MSC:
81V55Applications of quantum theory to molecular physics
81Q05Closed and approximate solutions to quantum-mechanical equations
81T80Simulation and numerical modelling (quantum field theory)
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[102]Simos T.E.: Stabilization of a four-step exponentially-fitted method and its application to the Schrödinger equation. Int. J. Mod. Phys C 18(3), 315–328 (2007) · Zbl 1200.65061 · doi:10.1142/S0129183107009261
[103]Simos T.E.: P-stability, trigonometric-fitting and the numerical solution of the radial Schrödinger equation. Comput. Phys. Commun. 180(7), 1072–1085 (2009) · Zbl 1198.81103 · doi:10.1016/j.cpc.2008.12.029
[104]Panopoulos G.A., Anastassi Z.A., Simos T.E.: Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions. J. Math. Chem. 46(2), 604–620 (2009) · Zbl 1198.81099 · doi:10.1007/s10910-008-9506-0
[105]Psihoyios G., Simos T.E.: Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math. 158(1), 135–144 (2003) · Zbl 1027.65095 · doi:10.1016/S0377-0427(03)00481-3
[106]Sakas D.P., Simos T.E.: Trigonometrically-fitted multiderivative methods for the numerical solution of the radial Schrödinger equation. MATCH Commun. Math. Comput. Chem. 53(2), 299–320 (2005)
[107]Psihoyios G., Simos T.E.: A family of fifth algebraic order trigonometrically fitted P-C schemes for the numerical solution of the radial Schrödinger equation. MATCH Commun. Math. Comput. Chem. 53(2), 321–344 (2005)
[108]Simos T.E.: Multiderivative methods for the numerical solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 50, 7–26 (2004)
[109]Van de Vyver H.: Efficient one-step methods for the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 711–732 (2008)
[110]Simos T.E.: Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004) · Zbl 1062.65075 · doi:10.1016/S0893-9659(04)90133-4
[111]Stavroyiannis S., Simos T.E.: 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) · Zbl 1169.65324 · doi:10.1016/j.apnum.2009.05.004
[112]Stavroyiannis S., Simos T.E.: A nonlinear explicit two-step fourth algebraic order method of order infinity for linear periodic initial value problems. Comput. Phys. Commun. 181(8), 1362–1368 (2010) · Zbl 1219.65067 · doi:10.1016/j.cpc.2010.04.002
[113]Anastassi Z.A., Simos T.E.: Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Reports 482, 1–240 (2009) · doi:10.1016/j.physrep.2009.07.005
[114]Vujasin R., Sencanski M., Radic-Peric J., Peric M.: A comparison of various variational approaches for solving the one-dimensional vibrational Schrödinger equation. MATCH Commun. Math. Comput. Chem. 63(2), 363–378 (2010)
[115]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
[116]Ixaru L.Gr., Rizea M.: Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985) · Zbl 0679.65053 · doi:10.1016/0010-4655(85)90100-6
[117]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
[118]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
[119]Simos T.E., Psihoyios G., Anastassi Z.: Preface, proceedings of the international conference of computational methods in sciences and engineering 2005. Math. Comput. Model 51(3–4), 137–137 (2010) · Zbl 1190.65007 · doi:10.1016/j.mcm.2009.08.004
[120]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
[121]T.E. Simos, G. Psihoyios, Special issue–selected papers of the international conference on computational methods in sciences and engineering (ICCMSE 2003) Kastoria, Greece, 12–16 September 2003–Preface. J. Comput. Appl. Math. 175(1), IX–IX (2005)
[122]T.E. Simos, J. Vigo-Aguiar, 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)
[123]T.E. Simos, Ch. Tsitouras, I. Gutman, Preface for the special issue numerical methods in chemistry. MATCH Commun. Math. Comput. Chem. 60(3) (2008)
[124]Simos T.E., Gutman I.: Papers presented on the international conference on computational methods in sciences and engineering (Castoria, Greece, September 12–16, 2003). MATCH Commun. Math. Comput. Chem 53(2), A3–A4 (2005)