| 成越,汤东升.二维复值金兹堡-朗道方程有限差分方法的无条件逐点最优误差估计[J].数学研究及应用,2024,44(2):248~268 |
| 二维复值金兹堡-朗道方程有限差分方法的无条件逐点最优误差估计 |
| Unconditional and Optimal Pointwise Error Estimates of Finite Difference Methods for the Two-Dimensional Complex Ginzburg-Landau Equation |
| 投稿时间：2023-03-16 修订日期：2023-07-08 |
| DOI：10.3770/j.issn:2095-2651.2024.02.011 |
| 中文关键词: 复值金兹堡-朗道方程 有限差分方法 无条件收敛 最优估计 逐点误差估计 |
| 英文关键词:complex Ginzburg-Landau equation finite difference method unconditional convergence optimal estimates pointwise error estimates |
| 基金项目:国家自然科学基金(Grant No.11571181), 南通大学科研启动基金(Grant No.135423602051). |
|
| 摘要点击次数: 113 |
| 全文下载次数: 137 |
| 中文摘要: |
| 本文对二维金兹堡-朗道方程的线性化和非线性Crank-Nicolson型的有限差分格式提出了改进的误差估计.对于线性化的Crank-Nicolson格式,本文运用数学归纳法得到了离散的$L^2$和$H^1$范数意义下的无条件误差估计.然而,这对于非线性格式不适用.因此,基于一个``截断''函数和能量分析方法,本文得到了非线性格式的无条件$L^2$和$H^1$误差估计,以及数值解的有界性. 此外,如果提高对精确解的假设,并借助能量分析方法和一些Sobleve不等式,可以得到无条件逐点最优误差估计.最后,本文给出一些数值例子来验证理论分析. |
| 英文摘要: |
| In this paper, we give improved error estimates for linearized and nonlinear Crank-Nicolson type finite difference schemes of Ginzburg-Landau equation in two dimensions. For linearized Crank-Nicolson scheme, we use mathematical induction to get unconditional error estimates in discrete $L^2$ and $H^1$ norm. However, it is not applicable for the nonlinear scheme. Thus, based on a `cut-off' function and energy analysis method, we get unconditional $L^2$ and $H^1$ error estimates for the nonlinear scheme, as well as boundedness of numerical solutions. In addition, if the assumption for exact solutions is improved compared to before, unconditional and optimal pointwise error estimates can be obtained by energy analysis method and several Sobolev inequalities. Finally, some numerical examples are given to verify our theoretical analysis. |
| 查看全文 查看/发表评论 下载PDF阅读器 |
|
|
|
