A Family of Amplified $3$-Step St{\"o}rmer Methods for Solving the Schr{\"o}dinger Equation and its Related Problems
نویسندگان
1 Department of Mathematics, The University of Alabama, Alabama, USA
2 Department of Mathematics, National University of Skills (NUS), Tehran, Iran
3 Department of Mathematics, The University of Tabriz, Tabriz, Iran
4 Department of Computational and Information Sciences, Stillman College, Alabama, USA
5 Department of Mathematics, Faculty of Basic science, The Khatam-ol-Anbia (PBU) University, Tehran, Iran
6 Department of Mathematics, Faculty of Mathematical Science, University of Maragheh, Maragheh, Iran
doi
10.22052/ijmc.2025.255596.1915چکیده
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.
