广义BBM-KdV方程的一个守恒C-N差分格式
A Conservative C-N Difference Scheme for the Generalized BBM-KdV Equation
DOI: 10.12677/PM.2021.114055, PDF,  被引量    国家自然科学基金支持
作者: 何 丽, 王 希, 胡劲松*：西华大学理学院，四川 成都
关键词: 广义BBM-KdV方程；差分格式；守恒；收敛性；稳定性；Generalized BBM-KdV Equation； Difference Scheme； Conservation； Convergence； Stability
摘要: 在进行非线性扩散波的研究时，BBM-KdV方程因能描述大量的物理现象如浅水波和离子波等而占有重要的地位，其数值研究少有涉及。本文研究了一类带有齐次边界条件的广义BBM-KdV方程的初边值问题，提出了一个具有二阶理论精度的两层非线性有限差分格式，合理模拟了问题本身的一个守恒量，并给出差分格式的先验估计，讨论其差分解的存在唯一性，并用离散泛函分析方法证明该格式的收敛性和无条件稳定性，最后通过数值模拟验证了该数值方法的可靠性。
Abstract: In the study of nonlinear diffusion waves, the BBM-KdV equation occupies an important position because it can describe a large number of physical phenomena such as shallow water waves and ion waves, and its numerical research is rarely involved. This paper studies the initial-boundary value problem of a generalized BBM-KdV equation with homogeneous boundary conditions, and proposes a two-level nonlinear finite difference scheme with second-order theoretical accuracy, which reasonably simulates a conserved quantity of the problem itself. A priori estimation of the difference scheme is given, and the existence and uniqueness of the difference decomposition is discussed. Discrete functional analysis is used to prove the convergence and unconditional stability of the scheme. Finally, the reliability of the numerical method is verified by numerical simulation.
文章引用：何丽, 王希, 胡劲松. 广义BBM-KdV方程的一个守恒C-N差分格式[J]. 理论数学, 2021, 11(4): 428-435. https://doi.org/10.12677/PM.2021.114055

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