A new phase-fitted modified Runge–Kutta pair for the numerical solution of the radial Schrödinger equation

Two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems

“…In Table II we show the quantitative properties of the higher order formula of the new embedded pair ARK6(4) versus the embedded pairs ADP5(4) derived in [14], PHADP5(4) derived in [13] and MPHADP5(4) derived in [18]. Since the pairs ADP5(4), PHADP5(4) and MPHADP5(4) are adapted versions of the classical pair Dopri5(4), they have kPTLEð0Þk 2 ¼ 3:99 Â 10 À4 .…”

“…The purpose of this paper is the design and construction of efficient embedded pairs of explicit adapted RK methods for solving oscillatory IVPs which are not adapted versions of classical RK algorithms from the literature as occurs in references [10,13,14,18]. The paper is organized as follows: Section 2 is devoted to present the basic concepts and results on adapted RK methods as well as the notation to be used in the rest of the paper.…”

“…Subsequently, Franco [10] analyzed the necessary and sufficient order conditions for this class of RK methods and he constructed explicit RK algorithms (up to order 4) as well as embedded pairs of explicit RK methods with orders 4 and 3. More recently, some authors [13,14,18] have proposed adapted versions of the http://dx.doi.org/10.1016/j.amc.2014. 11.097 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.…”

“…classical pair Dopri5(4) of Dormand and Prince [20] for solving oscillatory IVPs. These versions of the classical pair Dopri5 (4) are known as embedded pairs of adapted RK methods in [14] and embedded pairs of phase-fitted RK methods in [13,18]. In practical applications, it has been shown that phase-fitted and adapted RK methods are more accurate and efficient than non-fitted ones provided that the main frequency of the problem or a good approximation of it is known in advance (see the numerical results presented in [13,14,18].…”

“…Since the analytical solutions of these IVPs are usually not available, they can be solved by using general purpose numerical methods or by using codes specially adapted to the oscillatory behavior of their solutions. The design and construction of RK methods specially adapted to the numerical solution of oscillatory IVPs (1) has been considered by several authors (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein). The coefficients of these methods usually depend on the parameter m ¼ x h, where h is the integration step-size and x represents an approximation of the main frequency of the problem.…”