Phase properties of high order, almostP-stable formulae

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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“…[4], [5] We call phase-lag of an algorithm (2) with the related characteristic equation (7) the dominant term of the expression…”

Section: Definition 5 If the Interval Of Periodicity Of An Algorithmmentioning

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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“…, [15] For any method corresponding to the characteristic equation (6) the phase-lag is defined as the leading term in the expansion of …”

Section: Definition 3[14]mentioning

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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“…Furthermore, our definition of inhomogeneous dispersion is completely analogous to the one used in the literature (see e.g. [7], [13]- [15]). …”

Section: 2) Dy(l) ()+"° Imentioning

confidence: 99%

Phase-Lag Analysis of Implicit Runge–Kutta Methods

Houwen¹,

Sommeijer²

1989

SIAM J. Numer. Anal.

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Abstract. We analyse the phase errors introduced by implicit Runge-Kuttu methods when a linear inhomogeneous test equatiotl is integrated. It is shown that the homogeneous phase errors dominate if long interval integrations are performed. Homogeneous dispersion relations for the special class of DIRK methods are derived and a few high-order dispersive DIRK methods are constructed. These methods are applied to systems of linear ditferential equations with oscillating solutions and compared with the "conventional" DIRK methods of NiHsett and Crouzeix.Key words. numerical analysis, ordinary differential equatiotlS, Runge-Kutta methods, periodic solutions AMS(MOS) subject classification. 65L05 l. Introduction. In this paper, special diagonally implicit Runge-Kutta (DIRK) methods will be constructed for integrating systems of OD Es of the formwhen we know in advance that the solution is oscillating. Analogously to a generally adopted approach in the phase-lag analysis of numerical methods for second-order equations with oscillating solutions, we use the equation (cf.as a test equation. Here w represents a natural (or eigen) frequency of the system and w,, represents the frequency of the forced solution component.In § 2, we start by deriving explicit expressions for the phase lag introduced by general, implicit Runge-Kutta (RK) methods. The phase lag is composed of two parts: the homogeneous phase lag corresponding to the eigenmodes in the solution, and the inhomogeneous phase lag corresponding to the forced solution component. We will show that in calculations over long intervals of integration, the homogeneous phase lag tends to increase linearly, whereas the inhomogeneous phase error is constant. For this reason, we concentrate on the reduction of homogeneous phase errors.In § 3, we introduce the concept of a qth order dispersive stability fimction, and we show that such a stability function generates Runge-Kutta methods that have homogeneous phase errors of order q.From § 4 on, we confine our considerations to DIRK methods. We first derive the (dispersion) relations specifying a qth order dispersive stability function (we remark that for explicit Runge-Kutta methods these relations can be found in [8]). It is shown that there exists a one-parameter family of m-stage, pth order consistent DIRK methods with homogeneous phase errors of order q=2(m-lp/2j). In § 5, the dispersion relations are solved for one-, two-, three-, and four-stage DIRK methods and the resulting stability functions are constructed. These functions are dispersive of order q = 2m. The two-stage stability function turns out to be identical with the stability function of the well-known DIRK method of NQlrsett (10].

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Phase properties of high order, almostP-stable formulae

Cited by 201 publications

References 9 publications

A New Algorithm for the Approximation of the Schrödinger Equation

A New Algorithm for the Approximation of the Schrödinger Equation

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

Phase-Lag Analysis of Implicit Runge–Kutta Methods

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