A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

Hao, Zhao-Peng; Sun, Zhi‐zhong; Cao, Wanrong

doi:10.1002/num.21925

Cited by 31 publications

(14 citation statements)

References 21 publications

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“…Hao et al [29] suggested a high-order finite difference method for the 2D complex GL equation. They proved that the proposed difference scheme is uniquely solvable and unconditionally convergent.…”

Section: An Introduction About Hoc-adi Methodsmentioning

confidence: 99%

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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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Section: An Introduction About Hoc-adi Methodsmentioning

confidence: 99%

“…The convergent order in maximum norm was two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme was proposed in [29].…”

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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“…To construct a compact difference scheme for solving (1.1)–(1.4), the following lemmas are needed.Lemma () Suppose

u (x) \in C^{6} [x_{j - 1}, x_{j + 1}]

, then we have

\begin{matrix} left \\ \frac{u (x_{j - 1}) - 2 u (x_{j}) + u (x_{j + 1})}{h^{2}} = \frac{1}{12} [u'' (x_{j - 1}) + 10 u'' (x_{j}) + u'' (x_{j + 1})] + \frac{h^{4}}{360} u^{(6)} ({normalξ}_{j}) \\ \frac{u (x_{j - 1}) - u (x_{j + 1})}{2 h} = \frac{1}{6} [u' (x_{j - 1}) + 4 u' (x_{j}) + u' (x_{j + 1})] + \frac{h^{4}}{120} u^{(6)} ({normalζ}_{j}) \end{matrix}

where

{normalξ}_{j}, {normalζ}_{j} \in (x_{j - 1}, x_{j + 1}

…”

Section: The Compact Difference Scheme and Computational Methodsmentioning

confidence: 99%

“…In [4, 12-14, 18-21, 30-36], many conservative finite difference schemes were used for various nonlinear wave equations. Recently, there has been growing interest in high-order compact (HOC) methods for solving partial differential equations (PDEs) [30][31][32][33][34][37][38][39][40][41][42][43][44]. Therefore, a mass-conservative and high-order compact finite difference scheme which can preserves the solution to the SRLW equation is needed.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of a conservative weighted compact difference scheme for the symmetric regularized long wave equations

2018

Numerical Methods Partial

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In this article, a new weighted and compact conservative difference scheme for the symmetric regularized long wave (SRLW) equations is considered. The new scheme is decoupled and linearized in practical computation, that is, at each time step only two tridiagonal systems of linear algebraic equations need to be solved. It is proved by the discrete energy method that the compact scheme is uniquely solvable, the convergence and stability of the difference scheme are obtained, and its numerical convergence order is O ( normalτ 2 + h 4 ) in the L ∞ ‐norm. Numerical experiment results show that the scheme is efficient and reliable.

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“…The numerical simulation of the model is playing an important role in understanding the behaviors of the equations in both non‐fractional and fractional cases. In the context of standard nonlinear GLE, various numerical methods have been developed, including spectral method , notably finite difference method etc. In the setting of fractional version, however, the numerical study is rather young and scarce.…”

Section: Introductionmentioning

confidence: 99%

A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

Hao

Sun

2016

Numerical Methods Partial

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The numerical solution for the one‐dimensional complex fractional Ginzburg–Landau equation is considered and a linearized high‐order accurate difference scheme is derived. The fractional centered difference formula, combining the compact technique, is applied to discretize fractional Laplacian, while Crank–Nicolson/leap‐frog scheme is used to deal with the temporal discretization. A rigorous analysis of the difference scheme is carried out by the discrete energy method. It is proved that the difference scheme is uniquely solvable and unconditionally convergent, in discrete maximum norm, with the convergence order of two in time and four in space, respectively. Numerical simulations are given to show the efficiency and accuracy of the scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 105–124, 2017

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A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

Cited by 31 publications

References 21 publications

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

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

Numerical study of a conservative weighted compact difference scheme for the symmetric regularized long wave equations

A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

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