A‎ ‎Family‎ ‎of‎ ‎Amplified‎ ‎$3$-Step‎ ‎St{\"o}rmer‎ ‎Methods‎ ‎for Solving the Schr{\"o}dinger Equation and its Related Problems

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎The University of Alabama‎, ‎Alabama‎, ‎USA

2 Department of Computational and Information Sciences‎, ‎Stillman College‎, ‎Alabama‎, ‎USA

3 Department of Mathematics‎, ‎Faculty of Basic science‎, ‎The Khatam-ol-Anbia (PBU) University‎, ‎Tehran‎, ‎Iran

4 Department of Mathematics‎, ‎Faculty of Mathematical Science‎, ‎University of Maragheh‎, ‎Maragheh‎, ‎Iran

5 Department of Mathematics‎, ‎National University of Skills (NUS)‎, ‎Tehran‎, ‎Iran

6 Department of Mathematics‎, ‎The University of Tabriz‎, ‎Tabriz‎, ‎Iran

10.22052/ijmc.2025.255596.1915

Abstract

‎A family of complex amplified St{\"o}rmer methods is studied for solving initial value problems of second-order differential equations with periodic or orbital solutions‎.
‎The new complex amplified St{\"o}rmer methods depend upon a parameter $w> 0$‎, ‎vanish its complex amplifier‎, ‎and integrate precisely algebraic polynomials‎.
‎We believe that each method category (St{\"o}rmer method is one of them) has its complex amplifier‎.
‎When finding the coefficients of the St{\"o}rmer methods‎, ‎if the imaginary and real parts of the complex amplifier‎, ‎if necessary‎, ‎their derivatives are equal to zero‎, ‎high-capability methods are obtained‎.
‎The principal local truncation errors of the new explicit St{\"o}rmer methods are addressed‎. ‎Their stability regions are depicted in a plane where the vertical axis is the problem frequency and the Horizontal axis is the method frequency‎.
‎A collection of numerical examples illustrates the success of the new family of complex amplified St{\"o}rmer methods in addressing the Schr{\"o}dinger equation and other related problems‎. ‎The advantage of the new methods is showcased by discussing their relevance to some issues in chemistry‎.

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References

[1] A. C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation, J. Comput. Phys. 6 (1970) 378–391, https://doi.org/10.1016/0021-9991(70)90037-9.
[2] K. Mu and 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 (2015) 1239–1256, https://doi.org/10.1007/s10910-015-0484-8.
[3] E. Stiefel and D. G. Bettis, Stabilization of Cowell’s method, Numer. Math. 13 (1969) 154–175, https://doi.org/10.1007/BF02163234.
[4] J. M. Franco and J. F. Palacian, High order adaptive methods of Nyström-Cowell type, J. Comput. Appl. Math. 81 (1997) 115–134, https://doi.org/10.1016/S0377-0427(97)00030-7.
[5] J. F. Frankena, Störmer-Cowell: straight, summed and split. An overview, J. Comput. Appl. Math. 62 (1995) 129–154.
[6] G. D. Quinlan and S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, Astron. J. 100 1694–1700.
[7] J. D. Lambert and I. A. Watson, Symmetric multistip methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976) 189–202.
[8] T. E. Simos, A new methodology for the development of efficient multistep methods for first-order IVPs with oscillating solutions, Mathematics 12 (2024) #504, https://doi.org/10.3390/math12040504.
[9] J. M. Franco, I. Gómez and L. Rández, Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order, Numer. Algorithms 26 (2001) 347–363, https://doi.org/10.1023/A:1016629706668.
[10] L. Kramarz, Stability of collocation methods for the numerical solution of y00 = f(x; y), BIT 20 (1980) 215–222, https://doi.org/10.1007/BF01933194.
[11] J. M. Franco, A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators, J. Comput. Appl. Math. 161 (2003) 283–293, https://doi.org/10.1016/j.cam.2003.03.002.
[12] H. Saadat, S. H. Hassan Kiyadeh, R. Goudarzi Karim and A. Safaie, Family of phase fitted 3-step second-order BDF methods for solving periodic and orbital quantum chemistry problems, J. Math. Chem. 62 (2024) 1223–1250, https://doi.org/10.1007/s10910-024-01619-3.
[13] H. Saadat, S. H. Hassan Kiyadeh, A. Safaie, R. Goudarzi Karim and F. Khodadosti, A new amplification-fitting approach in Newton-Cotes rules to tackling the high-frequency IVPs, Appl. Numer. Math. 207 (2025) 86–96, https://doi.org/10.1016/j.apnum.2024.08.024.
Iranian Journal of Mathematical Chemistry
Volume 16, Issue 4
In Progress
December 2025
Pages 313-336

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