New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

“…Simos and Vanden Berghe et al constructed EF/TF explicit RK methods using different approaches. Since then, several authors constructed EF/TF RK methods,() EF/TF RKN methods,() and two‐step hybrid methods. () Here, we will construct modified methods following both approaches.…”

Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations

“…For nonlinear Schrödinger equations, taking the frequency of the problem into account, several useful trigonometrically/exponentially-fitted methods are proposed (see, e.g., [26][27][28][29][30][31][32]). Some authors, such as González, et al [33] and Franco [13], are interested in numerical integrators of Runge-Kutta-Nyström type adapted to initial value problems of the form y 00 ðtÞ þ x 2 yðtÞ ¼ f yðtÞ ð Þ; t 2 ½t 0 ; T; yðt 0 Þ ¼ y 0 ; y 0 ðt 0 Þ ¼ y 0 0 ;…”

Trigonometric–fitted hybrid four–step methods of sixth order for solving

“…Our group led by Prof. T. E. Simos has dealt for long with methods solving IVPs. [1][2][3][4] Such as Runge-Kutta, [5][6][7][8] Runge-Kutta-Nyström, [9][10][11][12] Numerov-type, [13][14][15][16][17] finite differences, 18,19 or other multistep methods. 20,21 The methods of interest in the current paper are defined below,…”