Fourier Spectral Approximation Research Articles

Fourier spectral approximation to long-time behavior of three dimensional Ginzburg-Landau type equation*

In this paper, we consider a three dimensional Ginzburg–Landau type equation with a periodic initial value condition. A fully discrete Galerkin–Fourier spectral approximation scheme is constructed, and then the dynamical properties of the discrete system are analyzed. First, the existence and convergence of global attractors of the discrete system are proved by a priori estimates and error estimates of the discrete solution, and the numerical stability and convergence of the discrete scheme are proved. Furthermore, the long-time convergence and stability of the discrete scheme are proved.

Fourier spectral approximation to long-time behaviour of the derivative three-dimensional Ginzburg–Landau equation

In this paper, we consider a derivative Ginzburg–Landau equation with periodic initial-value condition in three-dimensional space. A fully discrete Galerkin–Fourier spectral approximation scheme is constructed, and then the dynamical behaviour of the discrete system is analysed. Firstly, the existence of global attractors A N τ of the discrete system are proved by a priori estimate of the discrete solution. Next, the convergence of approximate attractors is proved by error estimates of the discrete solution. Furthermore, the long-time convergence as N → ∞ and τ → 0 simultaneously as well as the numerical long-time stability of the discrete scheme are obtained.

Fourier Spectral Approximation to Long-time Behavior of Dissipative Generalized KdV-Burgers Equations

In this paper, we consider a generalized KdV-Burgers equation with periodic initial value condition. Firstly, a fully discrete Galerkin-Fourier spectral approximation scheme, which is a linear one, is constructed. Next, the dynamical properties of the discrete system are analyzed for the autonomous case. The existence and the convergence of the global attractors of the discrete system are obtained by a priori estimates and the error estimates of the discrete solution. The stability of the discrete scheme is also proved. Finally, the long-time stability and the convergence of the discrete scheme are proved for the nonautonomous case. All results in this paper are obtained without any restriction on the time step size.

Conservation properties of multisymplectic integrators

Recent results on the local and global properties of multisymplectic discretizations of Hamiltonian PDEs are discussed. We consider multisymplectic (MS) schemes based on Fourier spectral approximations and show that, in addition to a MS conservation law, conservation laws related to linear symmetries of the PDE are preserved exactly. We compare spectral integrators (MS versus non-symplectic) for the nonlinear Schrödinger (NLS) equation, focusing on their ability to preserve local conservation laws and global invariants, over long times. Using Lax-type nonlinear spectral diagnostics we find that the MS spectral method provides an improved resolution of complicated phase space structures.

On the preservation of phase space structure under multisymplectic discretization

In this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and Fourier spectral approximations. Multisymplectic (MS) schemes are developed for two benchmark nonlinear wave equations, the sine-Gordon and nonlinear Schrödinger equations. We examine the implications of preserving the MS structure under discretization on the numerical scheme’s ability to preserve phase space structure, as measured by the nonlinear spectrum of the governing equation. We find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws, dynamical invariants and complicated phase space structures.

Approximations for KdV equation as a Hamiltonian system

KdV equation is one example of infinite-dimensional Hamiltonian systems. In this paper, we consider a Fourier spectral approximation and a collocation approximation for the KdV equation with emphasis on its Hamiltonian nature. In particular, we analyze the relations between the Fourier spectral method and the collocation method and in this manner we naturally show that several conservation laws of the KdV equation can be preserved for the collocation method.

Three level fourier spectral approximations for fluid flow with low mach number

Fourier spectral approximation, combined with a second-order time differencing technique for fluid flow with low Mach number is considered in this paper. Its generalized stability and convergence are strictly proven.

Spectral-finite element method for solving three-dimensional unsteady Navier-Stokes equations

This paper is devoted to the combined Fourier spectral and finite element approximations of three-dimensional, semi-periodic, unsteady Navier-Stokes equations. Fourier spectral method and finite element method are employed in the periodic and non-periodic directions respectively. A class of fully discrete schemes are constructed with artificial compression. Strict error estimations are proved. The analysis shows also that the classical two-dimensional velocity-pressure elements can be readily extended to solving such three-dimensional semi-periodic problems, provided they satisfy the two-dimensional “inf-suf” condition.

The fourier-chebyshev spectral method for solving two-dimensional unsteady vorticity equations

We propose a mixed method for solving the two-dimensional unsteady vorticity equations by using the Fourier-spectral approximation in the periodic direction and Chebyshev-spectral approximation in the non-periodic direction. Some numerical results are given, which are compared with those of other methods. Stability of the scheme and the optimal rate of convergence are proved.

Semi-discrete Fourier spectral approximations of infinite dimensional Hamiltonian systems and conservation laws

In this paper, we discuss the semi-discrete Fourier spectral approximation for infinite dimensional. Hamiltonian systems and pay more attention to the Hamiltonian structure. In this manner we consider further its preservation of conservation laws of the original Hamiltonian systems.