An optimized Runge–Kutta method for the solution of orbital problems

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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“…If for the algorithm (2) we have the following roots σ i , i = 1(1)2 k of its related equation (7) for all v ∈ (0, v 2 0 ),: v) , and…”

Section: Definition 3 [6]mentioning

confidence: 99%

“…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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“…More specifically the last years: -Phase-fitted methods and numerical methods with minimal phase-lag of RungeKutta and Runge-Kutta Nyström type have been developed in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The research on this subject has as a scope the production of numerical methods of Runge-Kutta and Runge-Kutta Nyström type which have vanished the phase-lag and/or the amplification factor.…”

Section: Brief Presentation Of the Literature On The Subjectmentioning

confidence: 99%

High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation

Simos

2012

J Math Chem

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In this paper we study the connection between: (i) closed Newton-Cotes formulae of high order, (ii) trigonometrically-fitted and exponentially-fitted differential methods, (iii) symplectic integrators. Several one step symplectic integrators have been produced based on symplectic geometry during the last decades (see the relevant literature and the references here). However, the study of multistep symplectic integrators is very poor. In this paper we investigate the High Order Closed Newton-Cotes Formulae and we write them as symplectic multilayer structures. We develop trigonometrically-fitted and exponentially-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes in order to solve the resonance problem of the radial Schrödinger equation. Based on the theoretical and numerical results, conclusions on the efficiency of the new obtained methods are given.Highly Cited

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“…In this paper, we intend to propose a new TDRK method based on the idea of [1]. The rest of the paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, Anastassi, et al [1] designed an optimized RK method with zero phase-lag and its derivatives, which is tested to by very efficient for the radial Schrödinger equation. In this paper, we intend to propose a new TDRK method based on the idea of [1].…”

Section: Introductionmentioning

confidence: 99%

An optimized explicit TDRK method for solving oscillatory problems

Wang¹,

Sun²,

Sun³

2016

J. Math. Computer Sci.

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In this paper, a new optimized explicit two-derivative Runge-Kutta (TDRK) method with frequencydepending coefficients is proposed, which is derived by nullifying the dispersion, the dissipation, and the first derivative of the dispersion. The new method has algebraic order four and is dispersive of order five and dissipative of order four. In addition, the phase analysis of the new method is also presented. Numerical experiments are reported to show the efficiency of the new method.

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An optimized Runge–Kutta method for the solution of orbital problems

Cited by 172 publications

References 4 publications

A New Algorithm for the Approximation of the Schrödinger Equation

A New Algorithm for the Approximation of the Schrödinger Equation

High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation

An optimized explicit TDRK method for solving oscillatory problems

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