Numerical treatment of the Sine-Gordon equations via a new DQM based on cubic unified and extended trigonometric B-spline functions

IF 2.1 3区物理与天体物理 Q2 ACOUSTICS

Wave Motion Pub Date : 2024-09-06 DOI:10.1016/j.wavemoti.2024.103409

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引用次数: 0

Abstract

The purpose of this work is to propose a new composite scheme based on differential quadrature method (DQM) and modified cubic unified and extended trigonometric B-spline (CUETB-spline) functions to numerically approximate one-dimensional (1D) and two-dimensional (2D) Sine-Gordon Eqs. (SGEs). These functions are modified and then applied in DQM to determine the weighting coefficients (WCs) of spatial derivatives. Using the WCs in SGEs, we obtain systems of ordinary differential equations (ODEs) which is resolved by the five-stage and order four strong stability-preserving time-stepping Runge–Kutta (SSP-RK_5,4) scheme. This method's precision and consistency are validated through numerical approximations of the one-and two-dimensional problems, showing that the projected method outcomes are more accurate than existing ones as well as an incomparable agreement with the exact solutions is found. Besides, the rate of convergence (ROC) is performed numerically, which shows that the method is second-order convergent with respect to the space variable. The proposed method is straightforward and can effectively handle diverse problems. Dev-C++ 6.3 version is used for all calculations while Figs. are drawn by MATLAB 2015b.

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通过基于立方统一和扩展三角 B-样条函数的新 DQM 对正弦-戈登方程进行数值处理

本研究的目的是提出一种基于微分正交法（DQM）和修正立方统一扩展三角 B 样条函数（CUETB 样条函数）的新复合方案，以数值逼近一维（1D）和二维（2D）正弦-戈登方程（SGE）。这些函数经过修正后应用于 DQM，以确定空间导数的加权系数（WC）。利用 SGE 中的 WCs，我们得到了常微分方程（ODE）系统，并通过五级四阶强稳定性保留时间步进 Runge-Kutta (SSP-RK5,4) 方案加以解决。通过对一维和二维问题的数值逼近验证了该方法的精确性和一致性，结果表明预测方法的结果比现有方法更精确，而且与精确解的一致性无可比拟。此外，还对收敛率（ROC）进行了数值计算，结果表明该方法对空间变量具有二阶收敛性。所提出的方法简单明了，能有效处理各种问题。所有计算均使用 Dev-C++ 6.3 版本，而图则由 MATLAB 2015b 绘制。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

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.