Optimizing a class of linear multi-step methods for the approximate solution of the radial Schrödinger equation and related problems with respect to phase-lag

Simos, Theodore E.

doi:10.2478/s11534-011-0074-8

Cited by 23 publications

(4 citation statements)

References 26 publications

(30 reference statements)

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“…We mention here that in the literature the last decades there are several variable step procedures for the approximation of the solution for systems of Schrödinger type equations [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

Section: Error Estimationmentioning

confidence: 99%

A New Algorithm for the Approximation of the Schrödinger Equation

Lin

Simos

2016

Open Physics

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Abstract:In this paper a four stages twelfth algebraic order symmetric two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives is developed for the first time in the literature. For the new proposed method: (1) we will study the phase-lag analysis, (2) we will present the development of the new method, (3) the local truncation error (LTE) analysis will be studied. The analysis is based on a test problem which is the radial time independent Schrödinger equation, (4) the stability and the interval of periodicity analysis will be presented, (5) stepsize control technique will also be presented, (6) the examination of the accuracy and computational cost of the proposed algorithm which is based on the approximation of the Schrödinger equation.

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Section: Error Estimationmentioning

confidence: 99%

A New Algorithm for the Approximation of the Schrödinger Equation

Lin

Simos

2016

Open Physics

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“…The problem which we will solve for our numerical experiments is the numerical solution of the radial time independent Schrödinger equation (31) using the mentioned above Woods-Saxon potential (34). Since this problem belongs to the boundary value problems with infinite interval of integration, it is necessary for its approximate solution the infinite integration interval to be approximated by a finite one.…”

Section: The Radial Schrödinger Equation and The Resonance Problemmentioning

confidence: 99%

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

Alolyan

Simos

2015

J Math Chem

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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 Electronic supplementary material The online version of this article (123 J Math Chem developed method by using the application of the new method to the approximate solution of the resonance problem of the radial Schrödinger 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.

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“…Some illustrative examples are also described. It can be pointed out that the use of multi-stage methods is also common as an efficiency optimization tool to improve the computational performances in other fields like, for instance, its use to improve the efficiency of numerical integration [22,23]. A comparison of the proposed stability analysis technique could be performed from the discussion in [24,25], and some of the references therein, related to the Lyapunov-type stability method.…”

Section: Introductionmentioning

confidence: 99%

On Some Sufficiency-Type Stability and Linear State-Feedback Stabilization Conditions for a Class of Multirate Discrete-Time Systems

Sen

2018

Mathematics

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This paper presents and discusses the stability of a discrete multirate sampling system whose sets of sampling rates (or sampling periods) are the integer multiple of those operating on all the preceding substates. Each of such substates is associated with a particular sampling rate. The sufficiency-type stability conditions are derived based on simple conditions on the norm, spectral radius and numerical radius of the matrix of the dynamics of a system parameterized at the largest sampling period.

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Optimizing a class of linear multi-step methods for the approximate solution of the radial Schrödinger equation and related problems with respect to phase-lag

Cited by 23 publications

References 26 publications

A New Algorithm for the Approximation of the Schrödinger Equation

A New Algorithm for the Approximation of the Schrödinger Equation

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

On Some Sufficiency-Type Stability and Linear State-Feedback Stabilization Conditions for a Class of Multirate Discrete-Time Systems

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