Numerical simulations of the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.

Section: Introductionsupporting

confidence: 85%

“…In Table , we present the

L_{\infty}

Section: Numerical Experimentsmentioning

confidence: 90%

A = .5

h = .5

Δ t = .05

are chosen over the interval

[- 150, 250]

.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Numerical solutions of Fourth‐order FDM and HBVM(4,2) with

A = .5

β = \sqrt{1 + 2 A / 3}

x_{0} = 0

h = .5

Δ t = .05

, and

T = 225

, left: Fourth‐order FDM , right: HBVM(4,2) [Color figure can be viewed at http://wileyonlinelibrary.com]…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Single Solitary Wavementioning

confidence: 99%

See 3 more Smart Citations

High‐order energy‐preserving schemes for the improved Boussinesq equation

Yan

Zhang

Zhao

et al. 2018

A Fourier pseudospectral method for a generalized improved Boussinesq equation

Borluk

Muslu

2014

In this article, we propose a Fourier pseudospectral method for solving the generalized improved Boussinesq equation. We prove the convergence of the semi-discrete scheme in the energy space. For various power nonlinearities, we consider three test problems concerning the propagation of a single solitary wave, the interaction of two solitary waves and a solution that blows up in finite time. We compare our numerical results with those given in the literature in terms of numerical accuracy. The numerical comparisons show that the Fourier pseudospectral method provides highly accurate results.to describe a large number of nonlinear dispersive wave phenomena, such as propagation of long waves on the surface of shallow water in both directions, propagation of long waves in one dimensional nonlinear lattices and propagation of ion-sound waves in a uniform isotropic plasma [3]. Here, the independent variables x and t denote spatial coordinate and time, respectively, and the parameter α = ±1. In the case of α = −1, (1.1) is called as the good Boussinesq equation; whereas in the case of α = 1, it is known as the bad Boussinesq equation. In [4], Bogolubsky showed that the bad Boussinesq equation is unstable under short wave perturbation and then he

A meshless finite point method for the improved Boussinesq equation using stabilized moving least squares approximation and Richardson extrapolation

2022

A meshless finite point method (FPM) is developed in this paper for the numerical solution of the nonlinear improved Boussinesq equation. A time discrete technique is used to approximate time derivatives, and then a linearized procedure is presented to deal with the nonlinearity. To achieve stable convergence numerical results in space, the stabilized moving least squares approximation is used to obtain the shape function, and then the FPM is adopted to establish the linear system of discrete algebraic equations. To enhance the accuracy and convergence order in time, the Richardson extrapolation is finally incorporated into the FPM. Numerical results show that the FPM is fourth‐order accuracy in both space and time and can obtain highly accurate results in simulating the propagation of a single solitary wave, the interaction of two solitary waves, the solitary wave break‐up and the solution blow‐up phenomena.