Trigonometrically fitted high-order predictor–corrector method with phase-lag of order infinity for the numerical solution of radial Schrödinger equation

An efficient four‐step multiderivative method for the numerical solution of second‐order IVPs with oscillating solutions

“…We have used several multistep methods for the integration of the five test problems. These methods are The Numerov's method which is indicated as I. The exponentially fitted two‐step method developed by Raptis and Allison 20 which is indicated as II. The exponentially fitted four‐step method developed by Raptis 27 which is indicated as III. The eight‐step ninth algebraic order method developed by Quinlan and Tremaine 24 which is indicated as IV. The 10‐step 11th algebraic order method developed by Quinlan and Tremaine 24 which is indicated as V. The 12‐step 13th algebraic order method developed by Quinlan and Tremaine 24 which is indicated as VI. The eight‐step method with phase‐lag and its first derivative equal to zero obtained in Reference 2 which is indicated as VII. The eight‐step method with phase‐lag and its first and second derivative equal to zero obtained in Reference 3 which is indicated as VIII. The 10‐step method with phase‐lag and its first and second derivatives equal to zero obtained in Reference 5 which is indicated as IX. The 10‐step method with phase‐lag and its first, second, and third derivatives equal to zero obtained in Reference 5 which is indicated as X. The 10‐step predictor‐corrector method developed in Reference 14 which is indicated as XI. An exponentially fitted eight‐order method obtained in Reference 39 which is indicated as XII. The four‐step P‐stable Obrechkoff method with vanished phase‐lag and its first, second, third, fourth, and fifth derivatives obtained in Reference 19 which is indicated as XIII. The explicit eight‐step 10th order method developed in Reference 40 which is indicated as XIV. The new four‐step P‐stable multiderivative method with vanished phase‐lag and its first, second, third, fourth, and fifth derivatives obtained in this article which is indicated as New. …”

“…In the past decades, various classes of methods have been designed for solving Equation () numerically via a variety of methods such as Runge‐Kutta, linear multistep, predictor‐corrector, and trigonometric or exponential methods 1‐15 . One of the most important properties of the numerical methods for solving Equation () is the P‐stability property.…”

“…In References 1,16‐18,20‐23, 43‐45, we can find the Runge‐Kutta and Runge‐Kutta‐Nyström type methods based on the phase‐fitting technique. In References 1‐15,23,24 exponentially and trigonometrically fitted multistep methods are obtained. We know that the numerical methods of second‐order initial value problems are divided into two classes: methods with constant coefficients and those with coefficients depending on the frequency of the problem.…”

“…The new method is from the second class. The purpose of this article is to introduce more efficient methods for the numerical solution of second‐order initial value problems with highly oscillatory solutions (see, eg, References 5‐36,37). To be more specific, the main aim of this article is to develop a new four‐step predictor P‐stable method which contains phase‐lag and some of the derivatives therein are equal to zero.…”

A new efficient implicit four-step method with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

“…Computational methods involving a parameter proposed by Gautschi [12], Jain et al [14] and Steifel and Bettis [30] yield numerical solution of problems of class (1). Chawla and et al [7,8], Anantha krishnaiah [3], Shokri and et al [20,21,22,23,24], Dahlquist [9], Franco [10], Lambert and Watson [15], Simos and et al [25,26,27], Saldanha and Achar [19], Achar [1], and Daele and Vanden Berghe [31] have developed methods to solve problems of class (2). We have organized the paper as follows: In Section 2, we present the preliminary concepts that requisite for theory of the new methodology.…”