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

a b s t r a c tOur new linear symmetric semi-embedded predictor-corrector method (SEPCM) presented here is based on the multistep symmetric method of Quinlan and Tremaine (1990), with eight steps and eighth algebraic order and constructed to solve numerically the twodimensional Kepler problem. It can also be used to integrate related IVPs with oscillatory solutions for which the frequency is unknown. Firstly we present a SEPCM (see Panopoulos and Simos, 2013 [36,37]) in pair form. This form has the advantage that reduces the computational expense. From this form we construct a new symmetric eight-step method. The new scheme has constant coefficients and algebraic order ten. We tested the efficiency of our newly developed scheme against some well known methods from the literature. We measure the efficiency of the methods and conclude that the new scheme is the most efficient of all the compared methods and for all the problems solved.

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“…y(t) = 0.200179477536 cos(1.01 t) + 2.46946143 · 10 −4 cos(3.03 t) + 3.04014 · 10 −7 cos(5.05 t) + 3.74 · 10 −10 cos(7.07 t) + · · · (see [43]). …”

Section: Two-dimensional Kepler Problemunclassified

An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

Panopoulos

Simos

2015

Journal of Computational and Applied Mathematics

109

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“…Methods with higher number of steps sharing variable coefficients were considered by Simos et al in Panopoulos and Simos. ()…”

Section: Introductionmentioning

confidence: 99%

“…Methods with higher number of steps sharing variable coefficients were considered by Simos et al in Panopoulos and Simos. [29][30][31] Here, we will study the general case of methods 1 and 2 that attain sixth algebraic order.…”

mentioning

confidence: 99%

Explicit hybrid six–step, sixth order, fully symmetric methods for solving y ″ = f (x,y)

Fang

Liu

Hsu

et al. 2019

Math Methods in App Sciences

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A family of explicit, fully symmetric, sixth order, six‐step methods for the numerical solution of y′′ = f(x,y) is studied. This family wastes two function evaluations per step and can be derived through interpolation techniques. An interval of periodicity is possessed and the phase lag is of high order. Numerical instabilities usually present in such type of multistep methods were circumvented. We conclude with extended numerical tests over a set of problems justifying our effort of dealing with the new methods.

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“…In the present paper, we study the approximate solution of special second order periodic initial or boundary-value problems (for the research on this subject see in [1] - [29] and references therein). The special second order differential equation (without rst derivative):…”

Section: Introductionmentioning

confidence: 99%

Implicit three-step methods with phase-lag and its derivatives equal to zero for the numerical solution of the Schrödinger equation and related problems

Alolyan¹,

Simos²

2015

AIP Conference Proceedings

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Articles you may be interested inA linear four-step method with vanished phase-lag and its first and second derivatives for the numerical solution of periodic initial and/or boundary value problems AIP Conf. Proc. 1618, 1031 (2014); 10.1063/1.4897910 Six-step methods with vanished phase-lag and its first derivatives for the approximate solution of the Schrödinger equation and related problems AIP Conf. Proc. 1618, 1026 (2014); 10.1063/1.4897909Four-step hybrid type methods with vanished phase-lag and its derivatives for each level for the numerical solution of the Schrödinger equation and related problems AIP Conf. Abstract.A family of implicit three-step methods for the numerical solution of special second order periodic initial or boundary-value problems (without rst derivative in their mathematical model) is investigated in this paper. The construction of the methods of the present family is based on the request phase-lag and its derivatives to be equal to zero. The study of the methods of the present family included • the investigation on the local truncation error of the methods and its comparison with the local truncation error of the corresponding methods with constant coef cients • the investigation on the stability for the new produced methods (using scalar test equation with frequency different than the scalar test equation for the phase-lag analysis) • the application of the new constructed methods to the resonance problem of the radial time independent Schrödinger equation in order to show their ef ciency.

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

Cited by 122 publications

References 21 publications

An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

Explicit hybrid six–step, sixth order, fully symmetric methods for solving y ″ = f (x,y)

Implicit three-step methods with phase-lag and its derivatives equal to zero for the numerical solution of the Schrödinger equation and related problems

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