A Fast and Efficient Numerical Algorithm for the Nonlocal Conservative Swift–Hohenberg Equation

Jingying Wang,Shuying Zhai

doi:10.1155/2020/7012483

Jingying Wang, Shuying Zhai

Open Access

https://doi.org/10.1155/2020/7012483

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Abstract

In this paper, we consider a new Swift–Hohenberg equation, where the total mass of this model is conserved through a nonlocal Lagrange multiplier. Based on the operator splitting method and spectral method, a fast and efficient numerical algorithm is proposed. Three numerical examples in both two and three dimensions are provided to illustrate that the proposed algorithm is a practical, accurate, and efficient simulation tool for the nonlocal Swift–Hohenberg equation.

Highlights

Introduction e SwiftHohenberg model [1] was first introduced by Swift and Hohenberg in studies of Rayleigh–Benard convection and has become one of the paradigms of the nonlinear dynamical system leading to complex pattern formation
We aim to develop a fast explicit algorithm based on the operator splitting method [14, 16, 17]. e main idea of the method is to solve problem (4) through three parts: the linear homogeneous heat equation was first solved via the Fourier spectral method and the nonlinear part and the nonlocal equation were solved explicitly
We present a time-splitting spectral method to simulate the asymptotic behavior of the solution of equation (4). e proposed method is based on the operator splitting method for the time and spectral method for the space

Summary

Research Article

A Fast and Efficient Numerical Algorithm for the Nonlocal Conservative Swift–Hohenberg Equation. We consider a new Swift–Hohenberg equation, where the total mass of this model is conserved through a nonlocal Lagrange multiplier. Zhang and Yang [15] very recently proposed a conservative Swift–Hohenberg equation with a nonlocal Lagrange multiplier to cancel out the variation in mass without influencing the original energy. We present a time-splitting spectral method to simulate the asymptotic behavior of the solution of equation (4). E operator splitting method for the conservative Swift–Hohenberg equation is constructed as follows. Based on the theory of spectral method [20], the solution to equation (11) derived with periodic boundary conditions can be represented by the following band-limited Fourier series: FN− 1[φ􏽥(t)](x, y) ≔ PNφ􏼁(x, y, t). Fourier transform (FFT) may be used to calculate the Fourier coefficients {φ􏽥p,q(t)} from the discrete function values {φi,j(t)}: FN[φ(t)](p, q)

Cpr er

We take the simulation parameters

Conclusions