A family of eight-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation

An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

“…We have constructed a predictor-corrector (PC) pair form (19) for the numerical integration of second-order IVPs (1). In this pair form a part of the predictor method is contained in the corrector method.…”

“…The semi-embedded predictor-corrector pair form (19) has the advantage that reduces the computational expense if the additions on the factor  m−1 i=0 a * i y n+i , are done twice. From the form (19) we have developed a new eight-step symmetric semi-embedded predictor-corrector method (SEPCM) (30).…”

“…From the form (19) we have developed a new eight-step symmetric semi-embedded predictor-corrector method (SEPCM) (30).…”

“…(i) Methods with constant coefficients (see [9,7,[10][11][12][13]) and (ii) Methods with coefficients dependent on the frequency of the problem (phase-fitted methods, exponentially-fitted methods and trigonometrically-fitted methods for second-order ODEs) (see [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]). …”

A new hybrid two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems

“…-In [9][10][11][12][13][14] exponentially and trigonometrically fitted Runge-Kutta and Runge-Kutta Nyström methods are obtained. -Multistep phase-fitted methods and multistep methods with minimal phase-lag are developed in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. -Symplectic integrators are studied in .…”