2015
DOI: 10.1080/00207160.2015.1076569
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Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation

Abstract: The Schrödinger equation is a model for many physical processes in quantum physics. It is a singularly perturbed differential equation where the presence of the small reduced Planck's constant makes the classical numerical methods very costly and inefficient. We design two new schemes. The first scheme is the nonstandard finite volume method, whereby the perturbation term is approximated by nonstandard technique, the potential is approximated by its mean value on the cell and the complex dependent boundary con… Show more

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Cited by 9 publications
(13 citation statements)
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“…And the initial guess V 0 s (x) is chosen to satisfy the same assumption as that for in (12)- (13). For the problem (17)- (20), when l = 1, following the similar process in Step I, we can derive the same stability as that in (15)-(16) for 1 . Then, using (19), the classical results for the elliptic problem (20) and the imbedding theorem, there hold…”
Section: Rtd Structure and Schrödinger-possion Systemmentioning
confidence: 80%
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“…And the initial guess V 0 s (x) is chosen to satisfy the same assumption as that for in (12)- (13). For the problem (17)- (20), when l = 1, following the similar process in Step I, we can derive the same stability as that in (15)-(16) for 1 . Then, using (19), the classical results for the elliptic problem (20) and the imbedding theorem, there hold…”
Section: Rtd Structure and Schrödinger-possion Systemmentioning
confidence: 80%
“…Usually, in traditional finite difference methods, the high-order derivatives in Taylor's series (23)- (24) are directly neglected as the truncation error terms. Such as in constructing the second-order standard finite difference scheme with a uniform mesh, the terms (n) j (n ≥ 4) are overlooked and (2) j is approximated by a universal second-order difference quotient 1 h…”
Section: Discrete Schemesmentioning
confidence: 99%
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“…The purpose of this work was to design explicit nonstandard finite difference schemes for the FH-N system of equations in the limit ε → 0. The work is motivated by earlier literature where NSFD methods were found to be efficient in cases where oscillatory solutions or problems that develop shock-like steep fronts are investigated, [1]. In addition, the rich dynamics of the temporal model presented in [20] motivates the idea of efficient discrete models for the system.…”
Section: Resultsmentioning
confidence: 99%
“…Among the various numerical techniques such as classical finite difference, finite volume, adaptive mesh, finite element, and spectral method for solving ODEs and PDEs, NSFD schemes have been proved to be one of the most efficient approaches in recent years. The authors in [8] proposed a nonstandard finite volume method for the numerical solution of a singularly perturbed Schrödinger equation. They have shown that the proposed nonstandard finite volume method is capable of reducing the computational cost associated with most classical schemes.…”
Section: Introductionmentioning
confidence: 99%