A new four-step P-stable Obrechkoff method with vanished phase-lag and some of its derivatives 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. …”

“…First, we are going to study the local truncation of the new method in case of the Schrödinger equation. For this, we will study the following methods: The eight‐step ninth algebraic order method developed by Quinlan and Tremaine 24 which is indicated as QT9. The 10‐step 11th algebraic order method developed by Quinlan and Tremaine 24 which is indicated as QT11. The 12‐step 13th algebraic order method developed by Quinlan and Tremaine 24 which is indicated as QT13. The classical method () with constant coefficients which is indicated as CL. The method produced by Alolyan and Simos 2 which is indicated as PLD1. The method produced by Alolyan and Simos 3 which is indicated as PLD12. The eight‐step method developed in paragraph 3.1 of Reference 1 which is indicated as PLD123a. The eight‐step method developed in paragraph 3.2 of Reference 1 which is indicated as PLD123b. The eight‐step method developed in paragraph 3.3 of Reference 1 which is indicated as PLD123c. The two‐step method developed in section 3 of Reference 6 which is indicated as NM3SPS3DV. The method produced by Shokri et al 19 which is indicated as Sho. …”

“…Although many linear multistep methods have since been proposed, the earliest method to solve () numerically is the Numerov's method. Recently Shokri et al 12,13,16‐19 have developed Obrechkoff type methods of various algebraic orders. 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.…”

“…Recently Shokri et al 12,13,16‐19 have developed Obrechkoff type methods of various algebraic orders. 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.…”

“…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.…”