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Modified three-derivative Runge–Kutta (MTHDRK) methods for the numerical solution of the resonant state for the Schrödinger equation are investigated. Order conditions are presented and oscillation-fitting conditions are derived. Two practical fifth-order explicit MTHDRK methods are constructed and the error analysis is carried out for large energy. The numerical results are presented for the numerical solution of the Schrödinger equation to show the robustness of our new methods. [ABSTRACT FROM AUTHOR]

Two new optimized three-derivative Runge–Kutta type methods with vanishing phase-lag and its first derivative for the numerical integration of Schrödinger equation are derived in this paper. We present the error analysis in terms of the asymptotic expressions of the local errors. Numerical results are reported to show the efficiency and robustness of the new methods for the numerical integration of the Schrödinger equation with the Woods–Saxon potential. [ABSTRACT FROM AUTHOR]

In this paper, we present a new family of trigonometrically fitted two-step Obrechkoff methods for the numerical solution of one-dimensional Schrödinger equation. Error analysis is carried out and the asymptotic expressions of the local errors for large energies are discussed. The numerical results are reported in the case of the resonance problem which are confirmed by the error analysis. [ABSTRACT FROM AUTHOR]

A new embedded 4(3) pair of modified two-derivative Runge–Kutta (TDRK) methods with First Same As Last (FSAL) property for the numerical solution of the Schrödinger equation is constructed in this paper. Both the error analysis and phase properties indicate good accuracy of the new pair especially for large eigenvalues. An application to the well-known Lennard-Jones potential confirms the theory and shows that the new pair is more efficient than some high-quality Runge–Kutta(–Nyström) pairs in the literature. [ABSTRACT FROM AUTHOR]

A family of exponentially fitted two-step hybrid methods for the numerical integration of the Schrödinger equation is investigated. The asymptotic expressions of the local errors for large energies are presented. The stability of the new methods is analyzed. Numerical results are reported for the widely used Woods–Saxon potential to show the efficiency of the new methods. The error analysis is clearly tested by the resonance problem. [ABSTRACT FROM AUTHOR]

A family of modified two-derivative Runge–Kutta (MTDRK) methods for the integration of the Schrödinger equation are obtained. Two new three-stage and fifth order TDRK methods are derived. The numerical results in the integration of the radial Schrödinger equation with the Woods–Saxon potential are reported to show the high efficiency of our new methods. The results of the error analysis are illustrated by the resonance problem. [ABSTRACT FROM AUTHOR]

For the numerical integration of highly oscillatory problems, revised trigonometrically fitted two-step hybrid methods (RTFTSH) with equation dependent coefficients are considered. The local truncation errors, stability and phase properties of the new method are analyzed. A feature of the new type of the methods is that the errors in the internal stages are assumed to contribute to the accuracy of the update. A new revised method RTFTSH4 of algebraic order four and phase-lag order four is derived. Numerical experiments are reported to show that the new method RTFTSH4 is much more efficient and robust than the standard fourth order method STFTSH4. [ABSTRACT FROM AUTHOR]

Two exponentially fitted two-derivative Runge-Kutta pairs for the numerical integration of the Schrödinger equation are presented in this paper. The asymptotic expressions of the local errors for large energies are given. The numerical results in the integration of the radial Schrödinger equation with the Woods-Saxon potential and the Lennard-Jones potential show the high efficiency of our new methods when compared with some famous optimized codes in the literature. [ABSTRACT FROM AUTHOR]

This paper focuses on adapted two-derivative Runge-Kutta (TDRK) type methods for solving the Schrödinger equation. Two new TDRK methods are derived by nullifying their phase-lags and the first derivatives of the phase-lags. Error analysis is carried out by means of asymptotic expressions of the local errors. Numerical results are reported to show the efficiency and robustness of the new methods in comparison with some RK type methods specially tuned to the integration of the radial time-independent Schrödinger equation with the Woods-Saxon potential. [ABSTRACT FROM AUTHOR]

Two exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods of algebraic order four are derived. The asymptotic expressions of the local errors for large energies are obtained. The numerical results in the integration of the radial Schrödinger equation with the Woods-Saxon potential show the high efficiency of our new methods compared to some famous optimized codes in the literature. [ABSTRACT FROM AUTHOR]

In this work a new modified embedded 5(4) pair of explicit Runge-Kutta methods is developed for the numerical solution of the Schrödinger equation. We investigate the error of the new pair, based on the error analysis we apply the higher order method to the resonance problem, also we apply the new embedded pair to elastic scattering phase-shift problem. The applications show the efficiency of our new developed embedded pair and the higher order method. [ABSTRACT FROM AUTHOR]

The study of the long time conservation for numerical methods poses interesting and challenging questions from the point of view of geometric integration. In this paper, we analyze the long time energy and magnetic moment conservations of two-step symmetric methods for charged-particle dynamics. A two-step symmetric method is proposed and its long time behaviour is shown not only in a normal magnetic field but also in a strong magnetic field. The approaches to dealing with these two cases are based on the backward error analysis and modulated Fourier expansion, respectively. It is obtained from the analysis that the method has better long time conservations than the variational method which was researched recently in the literature. [ABSTRACT FROM AUTHOR]