A family of hybrid 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/s0377-0427(97)00188-x

Cited by 49 publications

(2 citation statements)

References 29 publications

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“…The category of equations with an oscillatory/periodic solution deserves extra consideration (see [1,2]). Significant effort has been put into studying the numerical solution to the above equation or system of equations during the past two decades (for examples, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], and the references therein). For a more in-depth look at the techniques used to solve (1) with solutions presenting oscillating behavior, refer to [3,8,18] and the references therein; Quinlan and Tremaine [10] as well as [6,7,19]; and so on.…”

Section: Introductionmentioning

confidence: 99%

A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

Simos

2024

Mathematics

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In this research, we provide a novel approach to the development of effective numerical algorithms for the solution of first-order IVPs. In particular, we detail the fundamental theory behind the development of the aforementioned approaches and show how it can be applied to the Adams–Bashforth approach in three steps. The stability of the new scheme is also analyzed. We compared the performance of our novel algorithm to that of established approaches and found it to be superior. Numerical experiments confirmed that, in comparison to standard approaches to the numerical solution of Initial Value Problems (IVPs), including oscillating solutions, our approach is significantly more effective.

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Section: Introductionmentioning

confidence: 99%

A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

Simos

2024

Mathematics

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“…Physical applications include a large variety of diffusive and plasma phenomena 1,2 as well as nonlinear wave propagation, 3 the dynamics of self-gravitating mass distributions, 4 and the diffusion in energy space of photons due to Compton scattering. 5,6 Integro-partial differential equations are usually solved by integrating forward in time from a given initial condition using a predictor-corrector algorithm 7,8 or a global relaxation method. 9,10 The convergence properties of such indirect methods are often difficult to predict in advance, and usually depend rather sensitively on both the governing equation and the nature of the initial conditions.…”

Section: Introductionmentioning

confidence: 99%

A new perturbative technique for solving integro-partial differential equations

Becker

1999

Journal of Mathematical Physics

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Integro-partial differential equations occur in many contexts in mathematical physics. Typical examples include time-dependent diffusion equations containing a parameter ͑e.g., the temperature͒ that depends on integrals of the unknown distribution function. The standard approach to solving the resulting nonlinear partial differential equation involves the use of predictor-corrector algorithms, which often require many iterations to achieve an acceptable level of convergence. In this paper we present an alternative procedure that allows us to separate a family of integro-partial differential equations into two related problems, namely ͑i͒ a perturbation equation for the temperature, and ͑ii͒ a linear partial differential equation for the distribution function. We demonstrate that the variation of the temperature can be determined by solving the perturbation equation before solving for the distribution function. Convergent results for the temperature are obtained by recasting the divergent perturbation expansion as a continued fraction. Once the temperature variation is determined, the self-consistent solution for the distribution function is obtained by solving the remaining, linear partial differential equation using standard techniques. The validity of the approach is confirmed by comparing the ͑input͒ continued-fraction temperature profile with the ͑output͒ temperature computed by integrating the resulting distribution function.

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Effective Numerical Approximation of Schrödinger type Equations through Multiderivative Exponentially‐fitted Schemes

Psihoyios

Simos

2004

Appl Numer Analy & Comput

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In this paper an exponentially-fitted multiderivative method is developed for the numerical integration of the Schrödinger equation. We call the method multi-derivative since it uses derivatives of orders two and four. An application to the resonance problem of the radial Schrödinger equation indicates that the new method is more efficient than the Numerov method and other known methods found in the literature.

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

Cited by 49 publications

References 29 publications

A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

A new perturbative technique for solving integro-partial differential equations

Effective Numerical Approximation of Schrödinger type Equations through Multiderivative Exponentially‐fitted Schemes

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