Explicit Four-step Method Research Articles

A New Explicit Singularly P-Stable Four-Step Method for the Numerical Solution of Second Order IVPs

In this paper, we introduce a new symmetric explicit four-step method with variable coefficients for the numerical solution of second-order linear periodic and oscillatory initial value problems of ordinary differential equations. For the first time in the literature, we generate an explicit method with the most important singularly P-stability property. The method is multiderivative and has algebraic order eight and infinite order of phase-lag. The numerical results for some chemical (e.g. orbit problems of Stiefel and Bettis) as well as quantum chemistry problems (i.e. systems of coupled differential equations) indicated that the new method is superior, efficient, accurate and stable.

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

A new Multistage high algebraic order four-step method is obtained in this paper. It is the first time in the literature that a method of this category is developed and has vanishing of the phase-lag and its first, second, third, fourth and fifth derivatives. We study this new method by investigating: (1) the development of the new method, i.e. the calculation of the coefficients of the method in order the phase-lag and its first, second, third, fourth and fifth derivatives of the phase-lag to be vanished, (2) the determination of the formula of the Local Truncation Error, (3) the comparative analysis of the Local Truncation Error (with this we mean the application of the new method and similar methods on a test problem and the analysis of their behavior), (4) the stability of the new method, by applying the new obtained method to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the results of this application i.e. by investigating the interval of periodicity of the new obtained method. We finally study the computational behavior the new developed method by using the application of the new method to the approximate solution of the resonance problem of the radial Schrodinger equation. We prove the effectiveness of the new obtained method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.

Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation

Based on an optimized explicit four-step method, a new hybrid high algebraic order four-step method is introduced in this paper. For this new hybrid method, we investigate the procedure of vanishing of the phase-lag and its first, second, third and fourth derivatives. More specifically, we investigate: (1) the construction of the new method, i.e. the computation of the coefficients of the method in order its phase-lag and first, second, third and fourth derivatives of the phase-lag to be eliminated, (2) the definition of the local truncation error, (3) the analysis of the local truncation error, (4) the stability (interval of periodicity) analysis (using scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis). Finally, we investigate computationally the new obtained method by applying it to the numerical solution of the resonance problem of the radial Schrodinger equation. The efficiency of the new developed method is tested comparing this method with well known methods of the literature but also using very recently developed methods.

A high algebraic order predictor–corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems

In this paper an eighth algebraic order predictor–corrector explicit four-step method is studied. The main scope of this paper is to study the consequences of (1) the vanishing of the phase-lag and its first, second, third and fourth derivatives and (2) the high algebraic order on the efficiency of the new developed method. A theoretical and computational study of the obtained method is also presented. More specifically, the theoretical study of the new predictor–corrector method consists of: Finally, the computational study of the new predictor–corrector method consists of the application of the new produced predictor–corrector explicit four-step method to the numerical solution of the resonance problem of the radial time independent Schrodinger equation.

A predictor–corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation

A predictor–corrector explicit four-step method of sixth algebraic order is investigated in this paper. More specifically, we investigate the results of the elimination of the phase-lag and its first, second and third derivatives on the efficiency of the proposed method. The resultant method is studied theoretically and computationally. The theoretical investigation of the new hybrid method consists of: (1) the construction of the new method, (2) the definition (calculation) of the local truncation error, (3) the comparative local truncation error analysis (with other known methods of the same form), (4) the stability analysis using scalar test equation with frequency different than the frequency of the phase-lag analysis. Finally, we will study computationally the new obtained method. This study is based on the application of the new produced predictor–corrector explicit four-step method to the approximate solution of the resonance problem of the radial time independent Schrodinger equation.

A new explicit four-step method with vanished phase-lag and its first and second derivatives

A new explicit four-step method with vanished phase-lag and its first and second derivatives

On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative

In the present paper, we will investigate a family of explicit four-step meth ods first introduced by Anastassi and Simos ( 1) for the case of vanishing of phase-lag and its first derivative. These methods are efficient for the numerical solution of the Schr ¨ odinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions. As we mentioned before the main scope of this paper is the study of the elimination of the phase-lag and its fi rst derivative of the family of the methods to be investigated. A comparative error and stability analysis will be presented f or the studied family of four-step explicit methods. The new obtained methods will finally be tested on the resonance problem of the Schrequation in order to examine their efficiency.

An explicit four-step method with vanished phase-lag and its first and second derivatives

A study on the vanishing of the phase-lag and its first and second derivatives for a family of explicit four-step methods first introduced by Anastassi and Simos (J Comput Appl Math 236:3880–3889, 2012) is presented in this paper. The methods investigated in this paper belongs to the category of methods with frequency dependent coefficients. For these methods we will investigate the procedure of vanishing of the phase-lag and its first and second derivatives. For the new proposed methods we will define the local truncation error and we will study an local truncation error analysis. Finally we will compare the results of the error analysis with other known methods of the literature. We will study also the stability analysis of the new proposed method. We will apply the new produced methods on the resonance problem of the Schrodinger equation in order to investigate their efficiency.

New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. Part I: Construction and theoretical analysis

In this paper we develop and study new high algebraic order multiderivative explicit four-step method with phase-lag and its first, second and third derivatives equal to zero. For the produced methods we investigate their errors and stability. Based on the above mentioned analysis we will arrive to some remarks and conclusions about their the efficiency to the numerical integration of the radial Schrodinger equation.

A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation

In this paper a generator of hybrid explicit four-step methods with minimal phase-lag is developed. The methods are of sixth algebraic order and have large intervals of periodicity. The coefficients of the methods are determined in order to have minimal phase-lag. The efficiency of the new methods is showed by their application to the Schrödinger equation and by their comparison with other well-known methods.

AN ACCURATE EXPONENTIALLY FITTED EXPLICIT FOUR-STEP METHOD FOR THE NUMERICAL SOLUTION OF THE RADIAL SCHRÖDINGER EQUATION

We present here an accurate exponentially fitted explicit four-step method for the numerical integration of the one-dimensional Schrödinger equation. The formula considered contains free parameters which are defined in order to integrate exponential functions. We note that this is the first explicit four-step exponentially fitted method in the literature. Numerical results also indicate that the new method is much more accurate than other well-known methods.

An explicit four-step phase-fitted method for the numerical integration of second-order initial-value problems

An explicit four-step method with phase-lag of infinite order is developed for the numerical integration of second-order initial-value problems. Extensive numerical testing indicates that this new method can be generally more efficient than other four-step methods.