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In this paper, a new explicit fourth-order Runge–Kutta method adapted to oscillatory problems with two frequencies (μ and ν) is developed. The new method can integrate exactly the different harmonic oscillators y ′ = i μ y and y ′ = i ν y simultaneously. Numerical stability and phase properties of the new method are analysed. Numerical experiments are conducted to illustrate the high accuracy and competence of our new method in comparison with some highly effective methods from the recent literature. [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]

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]

A new family of one-parameter equation dependent Runge-Kutta-Nyström (EDRKN) methods for the numerical solution of second-order differential equations are investigated. The coefficients of new three-stage EDRKN methods are obtained by nullifying up to appropriate order of moments of operators related to the internal and external stages. A fifth-order EDRKN method that is dispersive of order six and dissipative of order five and a fourth-order EDRKN method that is dispersive of order four and zero-dissipative are derived. Phase analysis shows that there exist no explicit EDRKN methods that are P-stable. Numerical experiments are reported to show the high accuracy and efficiency of the new EDRKN methods. [ABSTRACT FROM AUTHOR]

The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods. [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]