Numerical study of time-splitting and space–time adaptive wavelet scheme for Schrödinger equations

Zhang, R.; Zhang, Kai; Zhou, Yan

doi:10.1016/j.cam.2005.03.086

Cited by 2 publications

(2 citation statements)

References 8 publications

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“…Adaptive algorithms based on heuristic mesh selection criteria exist in the literature for various cases of NLS equations (1.1), cf. [2,24,45,52,55]. Usually the criteria used for the construction of adaptive algorithms in these cases are based on structural properties known for the exact solution.…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Katsaounis¹,

Kyza²

2018

SIAM J. Numer. Anal.

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Abstract. We provide a posteriori error estimates in the L ∞ ([0, T ]; L 2 (Ω))−norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank-Nicolson-type scheme introduced by Besse in [9]. The space discretization consists of finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. Our analysis includes as special cases the cubic and quintic nonlinear Schrödinger equations in one spatial dimension and the cubic nonlinear Schrödinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.

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Section: Introductionmentioning

confidence: 99%

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Katsaounis¹,

Kyza²

2018

SIAM J. Numer. Anal.

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“…Bao [22] also proposed the time-splitting Chebyshev-spectral method to preliminarily solve Schrödinger equation with nonzero far-field conditions. Moreover, Zhang [14,23] et al presented a timesplitting and space-time adaptive wavelet scheme to solve Schrödinger equation with zero far-field boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Time-Splitting Chebyshev-Spectral Method for the Schrödinger Equation in the Semiclassical Regime with Zero Far-Field Boundary Conditions

Ni$^{

2013

J. Basic Appl. Sci.

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Semiclassical limit of Schrödinger equation with zero far-field boundary conditions is investigated by the timesplitting Chebyshev-spectral method. The numerical results of the position density and current density are presented. The time-splitting Chebyshev-spectral method is based on Strang splitting method in time coupled with Chebyshevspectral approximation in space. Compared with the time-splitting Fourier-spectral method, the time-splitting Chebyshevspectral method is unnecessary to treat the wave function as periodic and holds the smoothness of the wave function. For different initial conditions and potential (e.g. constant potential and harmonic potential), extensive numerical test examples in one-dimension are studied. The numerical results are in good agreement with the weak limit solutions. It shows that the time-splitting Chebyshev-spectral method is effective in capturing-oscillatory solutions of the Schrödinger equation with zero far-field boundary conditions. In addition, the time-splitting Chebyshev-spectral method surpasses the traditional finite difference method in the meshing strategy due to the exponentially high-order accuracy of Chebyshev-spectral method.

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Numerical study of time-splitting and space–time adaptive wavelet scheme for Schrödinger equations

Cited by 2 publications

References 8 publications

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Time-Splitting Chebyshev-Spectral Method for the Schrödinger Equation in the Semiclassical Regime with Zero Far-Field Boundary Conditions

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