(1+2)
IF 2.1 3区 物理与天体物理 Q2 ACOUSTICS
Anisha Devi, Om Prakash Yadav
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引用次数: 0

摘要

本文使用 Galerkin 有限元法(FEM)中的高阶形状元素研究广义本杰明-博纳-马霍尼-伯格斯(gBBMB)方程的孤波解。众所周知,有限元法中的高阶元素能产生更好的解近似结果;然而,文献中对这些元素的研究较少。因此,在对 gBBMB 方程进行有限元分析时,我们考虑了拉格朗日二次形状函数。我们采用 Galerkin 有限元近似法来推导半离散解的先验误差估计值。对于全离散解,我们采用了 Crank-Nicolson 方法,为了处理非线性问题,我们使用了带有 Crank-Nicolson 外推法的预测器-校正器方案。此外,我们还利用能量法对时间进行了稳定性分析。在空间上,我们观察到 L2(Ω) 准则的 O(h3) 收敛性和 H1(Ω) 准则的 O(h2) 收敛性。此外,在时间方向的最大规范中也获得了最优的 O(Δt2) 收敛性。我们在一些一维和二维空间的数值示例中检验了理论结果,包括单孤波的频散以及双孤波和三孤波的相互作用。为了证明本方案的效率和有效性,我们计算了 L2 和 L∞ 规范误差,以及质量、动量和能量不变式。我们将所获得的结果与现有的文献结果进行了数值和图形比较。我们发现二次形状函数提高了质量、动量和能量不变式的精确度,同时也为所考虑的非线性问题的 Galerkin 近似算法带来了更高的收敛阶数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Higher order Galerkin finite element method for (1+2)-dimensional generalized Benjamin–Bona–Mahony–Burgers equation: A numerical investigation

In this article, we study solitary wave solutions of the generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation using higher-order shape elements in the Galerkin finite element method (FEM). Higher-order elements in FEMs are known to produce better results in solution approximations; however, these elements have received fewer studies in the literature. As a result, for the finite element analysis of the gBBMB equation, we consider Lagrange quadratic shape functions. We employ the Galerkin finite element approximation to derive a priori error estimates for semi-discrete solutions. For fully discrete solutions, we adopt the Crank–Nicolson approach, and to handle nonlinearity, we utilize a predictor–corrector scheme with Crank–Nicolson extrapolation. Additionally, we perform a stability analysis for time using the energy method. In the space, O(h3) convergence in L2(Ω) norm and O(h2) convergence in H1(Ω) norm are observed. Furthermore, an optimal O(Δt2) convergence in the maximum norm for the temporal direction is also obtained. We test the theoretical results on a few numerical examples in one- and two-dimensional spaces, including the dispersion of a single solitary wave and the interaction of double and triple solitary waves. To demonstrate the efficiency and effectiveness of the present scheme, we compute L2 and L normed errors, along with the mass, momentum, and energy invariants. The obtained results are compared with the existing literature findings both numerically and graphically. We find quadratic shape functions improve accuracy in mass, momentum, energy invariants and also give rise to a higher order of convergence for Galerkin approximations for the considered nonlinear problems.

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来源期刊
Wave Motion
Wave Motion 物理-力学
CiteScore
4.10
自引率
8.30%
发文量
118
审稿时长
3 months
期刊介绍: Wave Motion is devoted to the cross fertilization of ideas, and to stimulating interaction between workers in various research areas in which wave propagation phenomena play a dominant role. The description and analysis of wave propagation phenomena provides a unifying thread connecting diverse areas of engineering and the physical sciences such as acoustics, optics, geophysics, seismology, electromagnetic theory, solid and fluid mechanics. The journal publishes papers on analytical, numerical and experimental methods. Papers that address fundamentally new topics in wave phenomena or develop wave propagation methods for solving direct and inverse problems are of interest to the journal.
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