High‐order energy‐preserving schemes for the improved Boussinesq equation
High‐order energy‐preserving schemes for the improved Boussinesq equation
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DOI：
10.1002/num.22249
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发表时间：
2018-07
影响因子：
3.9
通讯作者：
Jinliang Yan;Zhiyue Zhang;Tengjin Zhao;D. Liang
中科院分区：
文献类型：
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作者：
Jinliang Yan;Zhiyue Zhang;Tengjin Zhao;D. Liang
本文针对改进的 Boussinesq 方程提出了一类高阶能量守恒方案。为了推导能量守恒方案，我们首先通过傅里叶伪谱方法对改进的 Boussinesq 方程进行离散化，从而得到有限维哈密顿系统。然后，利用哈密顿边值法求解所获得的半离散系统，哈密顿边值法是一类新发展的能量守恒方法。所提出的方案在空间上可以达到谱精度，在时间上可以分别达到二阶、四阶和六阶精度。此外，所提出的方案可以将离散质量和能量保存在机器精度范围内。此外，为了显示所提方法的效率和准确性，将所提方法与有限差分法和有限体积元法进行了比较。给出了关于单个孤立波的传播、两个孤立波的相互作用以及波分裂的几个数值实验的结果。
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.