Local Discontinuous Galerkin Method for Solving Compound Nonlinear KdV-Burgers Equations

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-03-04 DOI:10.1002/mma.10851

Abhilash Chand, Jugal Mohapatra

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Abstract

In this work, an efficient local discontinuous Galerkin scheme is applied to numerically solve the nonlinear compound KdV-Burgers equation. The numerical scheme utilizes a local discontinuous Galerkin discretization technique in the spatial direction coupled with a higher order strong-stability-preserving Runge–Kutta scheme in the temporal direction. The $L^{2}$ stability analysis of the implemented numerical scheme, along with a detailed error estimate for smooth solutions, are also established by carefully selecting the interface numerical fluxes. In addition, numerical simulations are carried out using several illustrative examples, and the results obtained are then compared with solutions acquired by the analytical $\exp (- Φ (ξ))$ -expansion method to validate the acceptable accuracy and plausibility of the proposed numerical technique. Also, both two-dimensional and three-dimensional graphical representations are presented to visually demonstrate the physical significance of the resulting traveling wave solutions.

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求解复合非线性KdV-Burgers方程的局部不连续伽辽金法

本文采用一种有效的局部不连续伽辽金格式对非线性复合KdV-Burgers方程进行了数值求解。数值格式在空间方向上采用局部不连续伽辽金离散化技术，在时间方向上采用高阶强稳定保持龙格-库塔格式。通过仔细选择界面数值通量，建立了所实现数值方案的l2 $$ {L}^2 $$稳定性分析，并对光滑解进行了详细的误差估计。此外，还用几个实例进行了数值模拟。并将所得结果与解析式exp（−Φ （ξ）） $$ \exp \left(-\Phi \left(\xi \right)\right) $$展开法的解进行了比较，验证了所提数值技术的可接受精度和合理性。此外，还提供了二维和三维图形表示，以直观地展示所得到的行波解的物理意义。

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.