High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

Tian, Zhenfu; Yu, Peixiang

doi:10.1016/j.cpc.2009.12.022

Cited by 33 publications

(23 citation statements)

References 26 publications

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“…Moreover, these results also demonstrate that its order of convergence in space can go up to 6. In addition, we can see from the comparisons shown in Figures 2-4 and Tables 1-4 that when a relatively fine mesh is adopted, the present wavelet solutions have a much better numerical accuracy than those obtained by many other existing numerical methods, including PR-ADI [6,18], SD-HOC [10,18], HOC-ADI [18,23,34], L-HOC-ADI [24], FONCECD [28], FOLCECD [28], SSFD [9], and L-ADI [24].…”

Section: Numerical Examplesmentioning

confidence: 83%

“…Example 1. Consider the two-dimensional linear Schrö-dinger equation with trapping potential [6,10,18,24] i…”

Section: Numerical Examplesmentioning

confidence: 99%

“…with the initial and boundary conditions extracted from its exact solution Peaceman-Rachford alternating direction implicit method (PR-ADI) [6,18], semidiscrete high-order compact scheme (SD-HOC) [10,18], high-order compact alternating direction implicit method (HOC-ADI) [18], and the linearized compact alternating direction implicit method (L-HOC-ADI) [24].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…To effectively evaluate the performance of numerical solutions, we introduce the error norms max and 2 , and the corresponding convergence rates in space max and 2 , which are, respectively, defined by [9,18,20,23] …”

Section: Numerical Examplesmentioning

confidence: 99%

See 3 more Smart Citations

A Space-Time Fully Decoupled Wavelet Galerkin Method for Solving Multidimensional Nonlinear Schrödinger Equations with Damping

Wang

Zhou

Liu

2017

Mathematical Problems in Engineering

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On the basis of sampling approximation for a function defined on a bounded interval by combining Coiflet-type wavelet expansion and technique of boundary extension, a space-time fully decoupled formulation is proposed to solve multidimensional Schrödinger equations with generalized nonlinearities and damping. By applying a wavelet Galerkin approach for spatial discretization, nonlinear Schrödinger equations are first transformed into a system of ordinary differential equations, in which all matrices are completely independent of time and never need to be recalculated in the time integration. Then, the classical fourth-order explicit Runge-Kutta method is used to solve the resulting semidiscretization system. By studying several widely considered test problems, results demonstrate that when a relatively fine mesh is adopted, the present wavelet algorithm has a much better computational accuracy and efficiency than many existing numerical methods, due to its higher order of convergence in space which can go up to 6.

show abstract

Section: Numerical Examplesmentioning

confidence: 83%

“…Example 1. Consider the two-dimensional linear Schrö-dinger equation with trapping potential [6,10,18,24] i…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

See 2 more Smart Citations

A Space-Time Fully Decoupled Wavelet Galerkin Method for Solving Multidimensional Nonlinear Schrödinger Equations with Damping

Wang

Zhou

Liu

2017

Mathematical Problems in Engineering

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show abstract

“…The high-order compact (HOC) methods, because of their highorder accuracy and smaller stencils, are gradually gaining popularity in computational fluid dynamics, computational acoustics and computational electromagnetic (see [24] and references therein).…”

Section: An Introduction About Hoc-adi Methodsmentioning

confidence: 99%

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

Shokri

Afshari

2015

Computer Physics Communications

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Convergence of compact ADI method for solving linear Schrödinger equations

Liao

Sun

Shi

et al. 2011

Numerical Methods Partial

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A compact ADI scheme of second‐order in time and fourth‐order in space is proposed for solving linear Schrödinger equations with periodic boundary conditions. By using the recently suggested discrete energy method, it is shown that the stable compact ADI method is unconditionally convergent in the maximum norm. Numerical experiments, including the comparisons with the second‐order ADI scheme and the time‐splitting Fourier pseudospectral method, are presented to support the theoretical results and show the effectiveness of our method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

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High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

Cited by 33 publications

References 26 publications

A Space-Time Fully Decoupled Wavelet Galerkin Method for Solving Multidimensional Nonlinear Schrödinger Equations with Damping

A Space-Time Fully Decoupled Wavelet Galerkin Method for Solving Multidimensional Nonlinear Schrödinger Equations with Damping

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

Convergence of compact ADI method for solving linear Schrödinger equations

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