【作者】 张珂；

【导师】 刘建康；

【作者基本信息】 山西大学 ， 计算数学， 2018， 硕士

【摘要】 随着控制理论的发展,带有阻尼项的波动方程初边值稳定化控制逐渐成为一个重要的理论研究内容.其研究结果广泛应用于我们的日常生活.然而由实际问题建模得出的波动方程组往往带有复杂阻尼项,针对不同阻尼项,其对应方程的求解方法也不尽相同,且利用单纯的公式推导很难求得方程的精确解.数值分析的出现,为我们提供了一种逐渐逼近精确解的可能性.因此,对带阻尼项的二维波动方程进行数值研究,在理论和实际应用中都具有重要意义.第一部分,本文对如下带有混合阻尼边界条件的二维波动方程初边值问题（?）先进行全离散,进而构建了方程的三层全离散隐格式.对于其内部结点,本文利用中心差分格式来建立,而对边界结点,则采用四点二阶格式.随后对差分格式展开定性分析,引用能量分析法给出先验估计式,在此基础上,验证了解的唯一存在性、稳定性和L₂范数意义下关于时间维度和空间维度上二阶收敛性.最后理论结果通过算例进行验证.第二部分,本文依据交替方向法对如下带有Robin型阻尼边界的二维波动方程初边值问题（?）构建新的差分格式—交替方向隐格式（ADI格式）,其格式与上一部分所建立的全离散隐格式相比,格式更为简洁,避免了冗长复杂的方程组,计算量大幅减小.交替方向差分格式的优势在于将二维问题变换到一维,简化了方程求解.同时充分利用先验估计式证明所建格式在L₂范数意义下关于空间1阶时间1.5阶收敛.最后理论结果通过算例进行验证.第三部分,研究角度转变到如下带内部时滞阻尼的二维波动方程（?）利用中心差分格式对时滞项进行处理,空间上建立全离散隐格式,通过算例验证了差分格式的收敛性,结果显示所建格式在时间维度和空间维度上二阶收敛。更多还原

【Abstract】 With the development of control theory,the wave equation with damping term and initial boundary value stabilization control has gradually become an important research content.The research results can be widely used in our daily life.However,the wave equation modeling with the practical problems is often with complex damping,for different damping,the corresponding equation solving methods are not same.And use the simple formula is difficult to find exact solution,numerical analysis provides a possibility of infinite approximation of exact solutions for us.Therefore,the numerical study of two-dimensional wave equation is of great significance in both theory and practical application.In the first part,this paper deals with the initial boundary value problem of twodimensional wave equation with mixed damping boundary,（?）a fully discrete scheme is constructed.For internal nodes,the central difference scheme is used to construct,and for boundary nodes,For qualitative analysis of the scheme,a priori estimate must be given by energy analysis method,and then the existence of the solution is verified.Stability and second-order convergence of space and time directions in the sense of L₂norm.Finally,the theoretical results are verified by an example.The second part,based on the alternating direction method,the initial boundary value problem of the two-dimensional wave equation with Robin type damping boundary,（?）.Alternating direction implicit format（ADI）is constructed.Compared with the full discrete implicit scheme in the first part,the proposed scheme is more concise.The advantage of alternating direction scheme is that the two-dimensional problem can be simplified to one dimension,and then only need to be solved by tridiagonal equations on the time layer.A priori estimator proves that the proposed scheme converges on the first order time 1.5 order of the space in the sense of L₂norm.Finally,the theoretical results are verified by an example.In the third part of the paper,we study the two-dimensional wave equation with internal delay damping from the transformation of boundary damping to internal damping with time delay.（?）.The above equation is established by the central difference scheme for the time-delay term,and the fully discrete three-layer implicit scheme is still established in space.Because it is not possible to find a suitable method for qualitative analysis,the correctness of the difference scheme is verified by a numerical example,and the results show that the difference scheme is correct.It shows the second order convergence under infinite dimensional norm.更多还原