High-order P-stable multistep methods

Franco, J. M.; Palacios, M.

doi:10.1016/0377-0427(90)90001-g

Cited by 171 publications

(14 citation statements)

References 6 publications

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“…There is a vast literature on the study of TSH methods, see for example [3,4,7,8,10,13,14,20,[23][24][25][26]32]. Traditionally, the order conditions for such methods are usually derived by expansions in Taylor series.…”

Section: Tsh Methodsmentioning

confidence: 99%

Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)

Vyver

2007

Journal of Computational and Applied Mathematics

120

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This paper provides a theoretical framework for a new type of phase-fitted and amplification-fitted two-step hybrid (FTSH) methods which is introduced by the author in [H. Van de Vyver, A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems, Internat. J. Modern Phys. C 17 (2006) 663-675]. The methods constitute a modification of dissipative two-step hybrid methods in the sense that two free parameters are added to eliminate the phase-lag and the amplification error. The methods are useful only when a good estimate of the frequency of the problem is known in advance. The parameters depend on the product of the estimated frequency and the stepsize. The algebraic order, zero-stability, stability and phase properties are examined. The theory is illustrated with sixth-order explicit FTSH methods. Numerical results carried out on an assortment of test problems show the relevance of the theory.

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Section: Tsh Methodsmentioning

confidence: 99%

Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)

Vyver

2007

Journal of Computational and Applied Mathematics

120

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“…Accordingly, we use (24), (8), and (36)-(38) to derive (31), (33), and (39)-(41) into (26), we obtain…”

Section: Error Analysis Of Collocation Methodsmentioning

confidence: 99%

“…In the first class of schemes, the coefficients depend on some known periods or frequencies of the solutions, including the exponential-fitted method, the trigonometrically-fitted method, and the linear multi-step method (see [1][2][3][4] and the references therein). In the second class of schemes, the coefficients are constants, such as the Runge-Kutta-Nyström method, the linear multi-step method, the hybrid method, the Störmer-Cowell method, and the prediction-correction method (see [3,[5][6][7][8][9][10] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Laguerre-Gauss collocation method for initial value problems of second order ODEs

Yan

Guo

2011

Appl. Math. Mech.-Engl. Ed.

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“…-The Phase-Fitted Method (Case 1) developed in [1], which is indicated as Method NMPF1 -The Phase-Fitted Method (Case 2) developed in [1], which is indicated as Method NMPF2 -The New Obtained Method developed in Section 3 (Case 2), which is indicated as Method NMC2 -The New Obtained Method developed in Section 3 (Case 1), which is indicated as Method NMC1 3 with the term classical we mean the method of Section 3 with constant coefficients The numerically calculated eigenenergies are compared with reference values 4 . In Figures 8 and 9, we present the maximum absolute error Err max = |log 10 (Err) | where…”

Section: Radial Schrödinger Equation -The Resonance Problemmentioning

confidence: 99%

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

Simos¹

2014

Appl. Math. Inf. Sci.

121

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Abstract:In the present paper, we will investigate a family of explicit four-step methods 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 Schrödinger 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 first derivative of the family of the methods to be investigated. A comparative error and stability analysis will be presented for the studied family of four-step explicit methods. The new obtained methods will finally be tested on the resonance problem of the Schrödinger equation in order to examine their efficiency.

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High-order P-stable multistep methods

Cited by 171 publications

References 6 publications

Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)

Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)

Laguerre-Gauss collocation method for initial value problems of second order ODEs

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

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