Burgers方程的有限元后验误差估计及其应用

发布时间：2018-10-29 21:51

【摘要】：Burgers方程作为流体力学领域最基本的偏微分方程之一有非常广泛的应用,它是在某些情形下可以求出精确解的非线性偏微分方程,但是Burgers方程的解可能在某些局部区域内正则性比较差(激波现象),这将导致数值求解很困难.因此Burgers方程的高效数值计算方法研究具有重要的理论意义和应用价值,而基于后验误差估计的自适应有限元方法对求解具有局部奇性解的问题是十分有用的.全文共分为五章,引言介绍了自适应有限元的研究背景、Burgers方程的研究背景与意义、Burgers方程的研究现状等.第二章介绍了文中用到的不等式、定理和涉及到的Sobolev空间、网格剖分管理策略等预备知识.第三章、第四章,我们分别基于Cole-hopf变换对一维、二维具有Dirichlet边界条件的Burgers问题进行处理,然后运用最小二乘有限元方法在时间、空间上对变换后的热传导方程进行离散,得到半离散格式和全离散格式,构造了后验误差估计子,然后用具体的算例加以验证.一维我们选取较大的雷诺数,通过与均匀网格下的误差进行对比,结果表明,在误差相近的情况下,自适应网格数更小,提高了计算效率.二维情形下,在不同时间节点下的区域进行自适应和均匀网格剖分,数值模拟结果表明,本文的理论是正确的,构造的数值方法是可行的.第五章对全文做了小结以及对未来工作的展望.
[Abstract]:As one of the most basic partial differential equations in fluid mechanics, Burgers equation is a nonlinear partial differential equation which can be solved in some cases. However, the solution of Burgers equation may have poor regularity (shock wave phenomenon) in some local regions, which will make it very difficult to solve numerically. Therefore, the study of efficient numerical methods for Burgers equations is of great theoretical significance and practical value. However, adaptive finite element method based on posteriori error estimation is very useful for solving problems with local singularities. The paper is divided into five chapters. The introduction introduces the research background of adaptive finite element, the research background and significance of Burgers equation, the research status of Burgers equation and so on. In the second chapter, we introduce the inequalities, theorems, Sobolev spaces, mesh division management strategies and so on. In chapter 3, chapter 4, we deal with the Burgers problem with Dirichlet boundary condition based on Cole-hopf transform, then discretize the transformed heat conduction equation in time and space by using the least square finite element method. A posteriori error estimator is constructed for semi-discrete schemes and fully discrete schemes, and then verified by a concrete example. In one dimension we select a large Reynolds number and compare it with the error of uniform mesh. The results show that the adaptive mesh number is smaller and the computational efficiency is improved when the error is similar. In two dimensional case, the adaptive and uniform mesh generation is carried out in different time nodes. The numerical simulation results show that the theory in this paper is correct and the constructed numerical method is feasible. The fifth chapter makes a summary of the full text and prospects for future work.
【学位授予单位】：西安理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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