Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions

Panopoulos, G. A.; Anastassi, Zacharias; Simos, T. E.

doi:10.1007/s10910-008-9506-0

Cited by 90 publications

(5 citation statements)

References 95 publications

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“…The above optimized implicit symmetric multistep method (38), has eight steps, tenth algebraic order, tenth order of phase-lag (see [7]) and interval of periodicity …”

Section: The Implicit Methods (Corrector)mentioning

confidence: 99%

A New Optimized Symmetric Embedded Predictor- Corrector Method (EPCM) for Initial-Value Problems with Oscillatory Solutions

Panopoulos¹,

Simos²,

Arabia³

2014

Appl. Math. Inf. Sci.

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122

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Abstract:In this work a new optimized symmetric eight-step embedded predictor-corrector method (EPCM) with minimal phase-lag and algebraic order ten is presented. The method is based on the symmetric multistep method of Quinlan-Tremaine [1], with eight steps and eighth algebraic order and is constructed to solve numerically IVPs with oscillatory solutions. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new optimized method with minimal phase-lag is noticeably most efficient of all the compared methods and for all the problems solved including the two-dimensional Kepler problem and the radial Schrödinger equation.

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“…The above optimized implicit symmetric multistep method (38), has eight steps, tenth algebraic order, tenth order of phase-lag (see [7]) and interval of periodicity …”

Section: The Implicit Methods (Corrector)mentioning

confidence: 99%

A New Optimized Symmetric Embedded Predictor- Corrector Method (EPCM) for Initial-Value Problems with Oscillatory Solutions

Panopoulos¹,

Simos²,

Arabia³

2014

Appl. Math. Inf. Sci.

Self Cite

122

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“…• The phase-fitted symmetric linear eight-step method developed by Panopoulos, Anastassi and Simos in [3].…”

Section: The Methodsmentioning

confidence: 99%

“…The numerical solution of the Schrödinger equation and related initial value problems with oscillating/periodic solutions has attracted the interest of many researchers during the last few decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this work we integrate the onedimensional time-independent Schrödinger equation, which is given by…”

Section: Introductionmentioning

confidence: 99%

A new symmetric linear eight-step method with fifth trigonometric order for the efficient integration of the Schrödinger equation

Anastassi

2011

Applied Mathematics Letters

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a b s t r a c tOn the basis of a classical symmetric eight-step method, an optimized method with fifth trigonometric order for the numerical solution of the Schrödinger equation is developed in this work. The local truncation error analysis of the method proves the decrease of the maximum power of the energy in relation to the corresponding classical method, which renders the method highly efficient. This is confirmed by comparing the method to other methods from the literature while integrating the equation. The superiority of the method is strengthened by the existence of a larger interval of periodicity of the new method in comparison to the corresponding classical method.

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“…-Multistep phase-fitted methods and multistep methods with minimal phase-lag are developed in [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. The research on this subject has as a scope the production of numerical nultistep methods of several type (linear, predictor-corrector, hybrid etc) which have vanished the phase-lag.…”

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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Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions

Cited by 90 publications

References 95 publications

A New Optimized Symmetric Embedded Predictor- Corrector Method (EPCM) for Initial-Value Problems with Oscillatory Solutions

A New Optimized Symmetric Embedded Predictor- Corrector Method (EPCM) for Initial-Value Problems with Oscillatory Solutions

A new symmetric linear eight-step method with fifth trigonometric order for the efficient integration 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

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