A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations

Li, Leonard Z.; Sun, Hai-Wei; Tam, Sik-Chung

doi:10.1016/j.cpc.2014.10.008

Cited by 21 publications

(14 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…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%

“…In Table 1, we list the error norms 2 of numerical solutions at time = 1 under Δ = 1/25000 achieved, respectively, from the proposed wavelet method, and the higher order compact alternating direction implicit method (HOC-ADI) [34].…”

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

View full text Add to dashboard Cite

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%

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

View full text Add to dashboard Cite

show abstract

“…The above CCD scheme (3.10)-(3.15) is sixth-order accurate [30,37]. Combining the third-order accuracy in time, the overall accuracy of u, v and w is O((∆t) 3 + (∆x) 6 + (∆y) 6 + (∆z) 6 ).…”

Section: And the Known Function Vector Ismentioning

confidence: 99%

“…According to fourier analysis, the CCD scheme possesses better spectral resolution than many other existing high-order schemes [31]. For the compact three-point stencils and high-order accuracy, it has been proved to be very efficient to solve different kinds of differential equations [32][33][34][35][36][37][38][39].…”

mentioning

confidence: 99%

A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

Pan

Yue

et al. 2020

Comp. Appl. Math.

View full text Add to dashboard Cite

In this paper, an efficient and high-order accuracy finite difference method is proposed for solving multidimensional nonlinear Burgers' equation. The third-order three stage Runge-Kutta total variation diminishing (TVD) scheme is employed for the time discretization, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. Our method is third-order accurate in time and sixth-order accurate in space. The CCD-TVD method treats the nonlinear term explicitly thus it is very efficient and easy to implement. In addition, we prove the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments including both two-dimensional and three-dimensional problems have been conducted to demonstrate the high efficiency and accuracy of the proposed method.

show abstract

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

Zhou

2021

Wavelet Numerical Method and Its Applications in Nonlinear Problems

View full text Add to dashboard Cite

A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations

Cited by 21 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

A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

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

Contact Info

Product

Resources

About