求解奇异摄动和界面问题的高效数值方法

发布时间：2018-01-27 15:55

本文关键词： 奇异摄动量身定做有限点方法一致收敛界面问题直接线法　出处：《清华大学》2016年博士论文　论文类型：学位论文

【摘要】：在众多科学与工程领域,我们经常需要数值求解带奇性的偏微分方程初边值问题,这类问题一直吸引着许多数学家和工程师的注意。本文中主要研究求解带小参数的抛物方程奇异摄动问题以及平面星形区域上的二阶椭圆方程界面问题的数值方法。本文首先提出了求解抛物方程奇异摄动问题的量身定做有限点方法。这类问题由于最高阶项前面带有小参数,所以会在部分区域边界上产生边界层或在区域内部产生内层。在边界层或内层里,问题的解变化非常迅速,想要捕捉到这种变化比较困难。我们的量身定做有限点方法的主要思想是运用局部约化方程的解作为基函数来构造求解原问题的数值格式。在本文中,首先对热传导问题,我们采用多项式函数作为基函数,重构出了传统的有限差分格式。然后我们设计了求解反应对流扩散问题的量身定做有限点格式。将我们的格式应用到Shishkin网格上,得到了最大模意义下的一致收敛误差估计。我们的数值算例验证了方法的有效性。本文还讨论了平面星形区域上的二阶椭圆方程界面问题的直接线法。首先,我们运用一个合适的坐标变换,将原星形区域上的界面问题转换成一个新坐标系下半无限长条形区域上的间断系数问题;然后通过有限元逼近,我们得到了该间断系数问题的一个半离散近似,它等价于一个常微分方程组边值问题;通过求解此边值问题,最终得到了原界面问题的半离散近似解。我们还给出了近似解的误差估计。数值结果显示我们的方法在不知道原问题奇性的情况下也能得到问题真解的高精度近似。
[Abstract]:In many fields of science and engineering, we often need to solve the initial boundary value problems of partial differential equations with singularity. This kind of problem has been attracting the attention of many mathematicians and engineers. In this paper, the numerical square for solving singular perturbation problem of parabolic equation with small parameters and the interface problem of second-order elliptic equation in plane star domain is studied. In this paper, a method of customizing finite points for solving singular perturbation problem of parabolic equations is presented. This kind of problem has a small parameter in front of the highest order term. Therefore, the boundary layer will be generated on some regional boundaries or the inner layer within the region. In the boundary layer or inner layer, the solution of the problem changes very quickly. It is difficult to capture this change. The main idea of our tailor-made finite point method is to use the solution of the local reduction equation as the basis function to construct the numerical scheme for solving the original problem. First of all, we use polynomial function as the basis function for the heat conduction problem. After reconstructing the traditional finite difference scheme, we design a custom-made finite point scheme for the reaction-convection-diffusion problem and apply our scheme to the Shishkin mesh. The uniformly convergent error estimates in the sense of maximum norm are obtained. Our numerical examples verify the validity of the method. In this paper, we also discuss the direct line method for the boundary problems of second-order elliptic equations in a plane star domain. By using a suitable coordinate transformation, the interface problem in the original star region is transformed into the discontinuous coefficient problem in the semi-infinite strip region in a new coordinate system. Then, by finite element approximation, we obtain the semi-discrete approximation of the discontinuous coefficient problem, which is equivalent to the boundary value problem of ordinary differential equations. By solving the boundary value problem. Finally, the semi-discrete approximate solution of the original interface problem is obtained. The error estimates of the approximate solution are also given. The numerical results show that our method can also obtain the exact approximate solution of the problem without knowing the singularity of the original problem. As if.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

【相似文献】