An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation
type: Item
instance of: scholarly article
title: An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation
zbMATH Open document ID: 1412.65095
author: Pengde Wang
author: Chengming Huang
published in: BIT
publication date: 2018-09-20T00:00:00Z
review text: The paper deals with the numerical solution of the fractional Ginzburg-Landau equation \[ \partial_t u + (\nu + i\eta)(-\Delta)^{\alpha/2} u + (\kappa+ i \zeta) |u|^2 u - \gamma u = 0, \] where \(\alpha\in (1,2]\) is the order of the Riesz fractional derivative. The problem is discretized w.r.t. time by the second-order backward difference formula combined with the explicit second-order Gear's extrapolation. The space discretization is carried out by a fourth-order fractional quasi-compact method. At each time step, only a linear system with a coefficient matrix independent of the time level is solved. The authors derive the convergence rate of order \(O(\tau^2 + h^4)\) without any restriction concerning the size of the mesh and time steps \(h\) and \(\tau\), respectively. Numerical tests are provided to confirm the accuracy and efficiency of the proposed scheme.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 35Q56
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 6940695
zbMATH Keywords: fractional Ginzburg-Landau equation
zbMATH Keywords: Riesz fractional derivative
zbMATH Keywords: implicit-explicit method
zbMATH Keywords: fractional compact scheme
zbMATH Keywords: convergence
reviewed by: Vit Dolejsi
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: s10543-018-0698-9
OpenAlex ID: W2793191661
cites work: The world of the complex Ginzburg-Landau equation
cites work: Implicit-Explicit Methods for Time-Dependent Partial Differential Equations
cites work: Fourier spectral methods for fractional-in-space reaction-diffusion equations
cites work: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative
cites work: Fourth Order Accurate Scheme for the Space Fractional Diffusion Equations
cites work: Fractional diffusion on bounded domains
cites work: High-order algorithms for Riesz derivative and their applications. II
cites work: High-order algorithms for Riesz derivative and their applications. III
cites work: Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints
cites work: Numerical Approximation of a Time Dependent, Nonlinear, Space‐Fractional Diffusion Equation
cites work: Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation
cites work: Solving Ordinary Differential Equations II
cites work: A linearized high-order difference scheme for the fractional Ginzburg-Landau equation
cites work: A fourth-order approximation of fractional derivatives with its applications
cites work: A three‐level linearized compact difference scheme for the Ginzburg–Landau equation
cites work: On the continuum limit for discrete NLS with long-range lattice interactions
cites work: Attractors and Inertial Manifolds for Finite-Difference Approximations of the Complex Ginzburg--Landau Equation
cites work: Dynamics of the 3-D fractional complex Ginzburg-Landau equation
cites work: Finite difference approximations for fractional advection-dispersion flow equations
cites work: Fractional generalization of the Ginzburg-Landau equation: an unconventional approach to critical phenomena in complex media
cites work: Coupled fractional nonlinear differential equations and exact Jacobian elliptic solutions for exciton-phonon dynamics
cites work: Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg-Landau equation
cites work: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
cites work: Well-posedness and dynamics for the fractional Ginzburg-Landau equation
cites work: Q4303980
cites work: On Tsertvadze's difference scheme for the Kuramoto-Tsuzuki equation
cites work: A second-order accurate numerical approximation for the fractional diffusion equation
cites work: Fractional dynamics of coupled oscillators with long-range interaction
cites work: A class of second order difference approximations for solving space fractional diffusion equations
cites work: Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative
cites work: A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations
cites work: Maximum-norm error analysis of a difference scheme for the space fractional CNLS
cites work: A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations
cites work: A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation
cites work: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations
cites work: An implicit midpoint difference scheme for the fractional Ginzburg-Landau equation
cites work: Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation
cites work: Analysis of some finite difference schemes for two-dimensional Ginzburg-Landau equation
cites work: Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations
cites work: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives
cites work: A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation
cites work: A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation
cites work: Quasi-compact finite difference schemes for space fractional diffusion equations
cites work: Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term
DOI: 10.1007/S10543-018-0698-9
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