Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations

Kong, Linghua; Duan, Yuhua; Wang, Lan; Yin, Xiuling

doi:10.1016/j.mcm.2011.11.036

Cited by 18 publications

(17 citation statements)

References 23 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: 69%

“…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%

“…Figure 4 plots the error norms of the present wavelet solutions with Δ = 1/5000 as a function of the number of grid points in space , as well as those obtained by using the split-step finite difference method (SSFD) [9], HOC-ADI [23,34], linearized alternating direction implicit scheme (L-ADI) [24], L-HOC-ADI [24], and CCD-PRADI [34]. Table 2.…”

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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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.

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Section: Numerical Examplesmentioning

confidence: 69%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

See 1 more Smart Citation

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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“…In fact, all the conclusions in the following sections can be easily extended to homogeneous problems with minor modifications. (17) and (18) Proof. Assuming that there exist (17) and (18).…”

Section: Numerical Analysis Of the Hoc Imentioning

confidence: 97%

“…It features spectral-like resolution, compactness and resource saving. It has been widely used in numerical solving partial differential equations [7,10,13,[17][18][19][20][21]. The basic idea of this method is to fully use the self information of the equation to be simulated.…”

Section: Proposition 1 If the Initial Valuesmentioning

confidence: 99%

Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations

Kong

Hong

et al. 2015

Numerical Methods Partial

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In the manuscript, we present several numerical schemes to approximate the coupled nonlinear Schrödinger equations. Three of them are high-order compact and conservative, and the other two are noncompact but conservative. After some numerical analysis, we can find that the schemes are uniquely solvable and convergent. All of them are conservative and stable. By calculating the complexity, we can find that the compact schemes have the same computational cost with the noncompact ones. Numerical illustrations support our analysis. They verify that compact schemes are more efficient than noncompact ones from computation cost and accuracy.

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Space–Time Fully Decoupled Wavelet-Based Solution to Nonlinear Problems

Zhou

2021

Wavelet Numerical Method and Its Applications in Nonlinear Problems

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Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations

Cited by 18 publications

References 23 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

Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations

Space–Time Fully Decoupled Wavelet-Based Solution to Nonlinear Problems

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