A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

Thomas, R. M.; Simos, T. E.; Mitsou, G.V.

doi:10.1016/0377-0427(94)00126-x

Cited by 33 publications

(9 citation statements)

References 21 publications

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“…In [32] Thomas, Simos and Mitsou have developed a family of Numerov-type exponentially fitted hybrid methods of the formȳ n = y n − ah 2 (y n+1 − 2y n + y n−1 ), y n = y n − bh 2 (y n+1 − 2ȳ n + y n−1 ),…”

Section: B4 Additional Implicit Hybrid Methodsmentioning

confidence: 99%

“…In [31] Psihoyios and Simos have constructed a trigonometrically fitted predictor-corrector method with trigonometric order two for the integration of first order ODEs. In [32] Thomas, Simos and Mitsou have developed a family of Numerov-type exponentially fitted hybrid methods. In [33] Psihoyios and Simos have constructed an implicit hybrid method.…”

Section: Introductionmentioning

confidence: 99%

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Numerical multistep methods for the efficient solution of quantum mechanics and related problems

Anastassi¹,

Simos²

2009

Physics Reports

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Section: B4 Additional Implicit Hybrid Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical multistep methods for the efficient solution of quantum mechanics and related problems

Anastassi¹,

Simos²

2009

Physics Reports

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“…However, some authors concluded that their ef-based methods are also Pstable; see, e.g., Jain et al, [37] and [38], Anantha Krishnaiah, [2], Denk, [16] and Thomas et al, [61]. The main reason isthat in their investigations wand k were assumed equal, and then only the situation along the line e = v has been examined.…”

Section: Linear Stability Theory and P-stabilitymentioning

confidence: 99%

Exponential Fitting

Ixaru¹,

Berghe²

2004

111

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Originally published by Kluwer Academic Publishers in 2004Softcover reprint ofthe hardcover lst edition 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Additional material to this book can be downloaded from http://extras.springer.com Contents Preface XI Acknowledgments xiii Subroutine REGSOLV2 71 4. NUMERICAL DIFFERENTIATION, QUADRATURE AND INTERPOLATION 77 1 Numerical differentiation 77 V Vl EXPONENT/AL FITTING PrefaceExponential fitting is a procedure for an efficient numerical approach offunctions consisting of weighted sums of exponential, trigonometric or hyperbolic functions with slowly varying weight functions. Operations on such functions Iike numerical differentiation, quadrature, interpolation or solving ordinary differential equations whose solution is of this type are of a real interest nowadays in many phenomena as oscillations, vibrations, rotations or wave propagation. The behaviour of quantum particles is also described by this type of functions.We witnessed the field for many years and contributed in it. Since the total number of papers accumulated so far in this field is over 200, and these papers are spread over journals with various profiles, to mention only those of applied mathematics, of computer science or of computational physics and chemistry, we thought that the time has come for a compact and systematic presentation of this vast material. lt is hoped that in this way many persons who are faced with such problems in their own activity would be better helped than searching for the needed information in such an abundant literature. This is what we do in this book which covers a series of aspects, ranging from the theory of the procedure up to direct applications and sometimes including ready-to-use programs.The writing of this book was decided two years ago. Since our working places are not close to each other, we agreed to share the effort: the first author has taken upon him Chapters 1 to 5, while the second author was responsible for Chapter 6. However, we both share equal responsibility for the whole text.We are indebted to a number of colleagues for discussions and the collaboration in several periods of our research:

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“…, xP exp(±vx)}. The methods by Coleman [3], Simos [11] and Thomas et al [12] are in this category. Runge-Kutta methods in the first or second categories are not yet found.…”

Section: Introductionmentioning

confidence: 99%

A functional fitting Runge-Kutta method with variable coefficients

Ozawa

2001

Japan J. Indust. Appl. Math.

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In this paper, we propose a functional fitting s-stage Runge-Kutta method which is based on the exact integration of the set of the linearly independent functions pt(t), (i = 1, ... , s). The method is exact when the solution of the ODE can be expressed as the linear combination of cpt(t), although the method has an error for general ODE. In this work we investigate the order of accuracy of the method for general ODEs, and show that the order of accuracy of the method is at least s, if the functions w(t) are sufficiently smooth and the method is non-confluent. Furthermore, it is shown that the attainable order of the method is 2s, like conventional Runge-Kutta methods. Two-and three-stage methods including embedded one of this type are developed.

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A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

Cited by 33 publications

References 21 publications

Numerical multistep methods for the efficient solution of quantum mechanics and related problems

Numerical multistep methods for the efficient solution of quantum mechanics and related problems

Exponential Fitting

A functional fitting Runge-Kutta method with variable coefficients

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