带有阻尼项的广义SRLW方程的有限差分逼近-应用数学专业论文.docxVIP

西华 西华大学硕士学位论文 带有 带有阻尼项的广义 SRLW 方程的有限差分逼近 I I 摘 要 在实际问题中，粘性阻尼是不可避免的，而且与色散一样起着十分重要的作用。由 于考虑了阻尼和耗散的影响，所以带有阻尼项的对称正则长波方程是反映非线性离子声 波运动本质现象的合理模型，对其数值解的研究也非常有意义。本文考虑了具有齐次边 界条件的一类带有阻尼项的广义对称正则长波方程的有限差分方法，构造了一个两层非 线性有限差分格式和一个三层线性有限差分格式，其中两层非线性有限差分格式合理地 模拟了初边值问题本身的一个守恒性质。分别给出了这两个格式的差分解的先验估计和 存在唯一性，并用能量方法分析了格式的二阶收敛性与无条件稳定性。数值算例表明， 我们的方法是可行的。 关键词：阻尼耗散；广义对称正则长波方程；差分格式；守恒；收敛性；稳定性； 数值实验 Abstract In applications, the viscous damping effect is inevitable and it plays the same important role as the dispersive effect. For the influence of damping and dissipation, generalized SRLWE are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered, and it is meaningful to research of its numerical solution. In this paper, a finite difference method for the damping dissipative generalized SRLWE with a homogeneous boundary conditions was considered. Both a nonlinear two-level conservative finite difference scheme and a linear three-level conservative finite difference scheme are designed. The nonlinear two-level conservative finite difference scheme simulate the conservation properties of the problem well. It is shown that the priori error estimate and uniquely solvable for the solutions of those finite difference schemes. It is proved that those finite difference schemes was a convergence of second-order convergence and unconditionally stable by the discrete energy method. Numerical examples confirm the theoretical results. Key Words：Damping Dissipation; Generalized Symmetric Regularized Long Wave Equations; Finite Difference Scheme; Conservation; Convergence; Stability; Numerical Experiment II 西华 西华大学硕士学位论文 III III 目 录 HYPERLINK \l _bookmark0 摘 要 I HYPERLINK \l _bookmark1 Abstract II HYPERLINK \l _bookmark2 0 引言 1 HYPERLINK \l _bookmark3 0.1 研究背景 1 HYPERLINK \l _bookmark4 0.2 预备知识及相关引理 2 HYPERLINK \l _bookmark5 1 两层非线性守恒有限差分格式 4 HYPERLINK \l _bookmark6 1.1 差分格式及其守恒律 4 HYPERLINK \l _bookmark7 1.2 差分解的先验估计 5 HYPERLIN