2013
DOI: 10.1186/1687-1847-2013-12
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Nonstandard finite difference variational integrators for nonlinear Schrödinger equation with variable coefficients

Abstract: In this paper, the idea of nonstandard finite difference discretization is employed to develop two variational integrators for the nonlinear Schrödinger equation with variable coefficients. These integrators are naturally multi-symplectic, and their multi-symplectic structures are presented by the multi-symplectic form formulas. Local truncation errors and convergences of the integrators are briefly discussed. The effectiveness and efficiency of the proposed schemes, such as the convergence order, numerical st… Show more

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Cited by 12 publications
(6 citation statements)
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“…The physical behavior of the numerical solutions described in Fig. 6 is coincident with that given in [39,40]. It can be seen that the Taylor approximation method of the CFDS-PIM has high accuracy.…”
Section: Numerical Examples and Discussionsupporting
confidence: 61%
“…The physical behavior of the numerical solutions described in Fig. 6 is coincident with that given in [39,40]. It can be seen that the Taylor approximation method of the CFDS-PIM has high accuracy.…”
Section: Numerical Examples and Discussionsupporting
confidence: 61%
“…The flexibility afforded by the NSFD scheme, in terms of its construction, means that the scheme secures consistency with the continuous pharmacokinetic models arising from compartmental or physiological models with respect to the different dynamical characteristics of the systems [ 17 ]. As a consequence, we were able to investigate the dynamics of the models and the impact of the parameter value choices employed.…”
Section: Discussionmentioning
confidence: 99%
“…The NSFD schemes developed by Mickens et al [ 12 , 13 , 14 , 15 , 16 ] were proposed to compensate for the weaknesses of methods such as the SFD methods; numerical instabilities being a prime example. As commented on by Liao and Ding [ 17 ], with regard to the positivity, boundedness, and monotonicity of solutions, NSFD schemes have performed better than SFD schemes. Because it is more flexible in its construction, an NSFD scheme can more easily preserve certain properties and structures obeyed by the original equations and can have better dynamical consistency for dynamical problems [ 17 ].…”
Section: Methodsmentioning
confidence: 95%
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“…We call it the simple box scheme to distinguish it from other Runge-Kutta box-based schemes. There are plenty of multisymplectic low-order methods applicable to Schrödinger's equation [26,14,10,22,9,5]. Most are based on box-like schemes and are second order.…”
Section: Introductionmentioning
confidence: 99%