边界层内层问题的稳定和高效的数值方法

发布时间：2018-03-29 20:13

本文选题：边界层　切入点：Oseen流　出处：《清华大学》2016年博士论文

【摘要】：在众多科学与工程领域的数学物理问题中,会出现各类边界层和内层现象,例如不可压流体中的粘性边界层,可压缩气体动力学中的内激波层,双曲型方程的初始层等。这类问题的一个重要特点是,在一个很窄的区域内,问题解本身或者其空间/时间导数会变化很快。传统的计算方法处理这类问题时其计算代价将极其昂贵。因此,人们往往需要结合其他技巧来设计更为有效的计算方法。本文将渐近分析等应用数学工具应用到时空上具有边界层/内层现象的偏微分方程的数值求解,得到了具有一致收敛性的高效数值方法。对于空间上有边界层和内层现象的偏微分方程,本文研究了大Reynolds数的Oseen流问题,多介质辐射扩散问题,以及源于玻色-爱因斯坦凝聚中的奇异摄动非线性特征值问题。Oseen流是不可压Navier-Stokes流的线性近似,我们通过方程分解技巧和人工边界方法将Oseen流问题转化为一个等价的有界计算区域上的二阶椭圆方程边值问题,再利用量身定做有限点方法进行数值求解。数值结果表明我们的方法关于Reynolds数是稳定的,且能在粗网格下捕捉到边界层和内层现象。非平衡辐射扩散方程组描述的是惯性约束聚变过程,我们提出了单调的量身定做有限点方法来进行数值求解。我们的方法具有保正性,数值结果表明我们的方法能够在粗网格下捕捉到陡锋的传播,而且能够推广到间断系数的情形。本文中的奇异摄动非线性特征值问题描述的是强相斥作用下的玻色-爱因斯坦凝聚现象,我们通过匹配渐近展开法给出了解在边界层和内层附近的渐近展开形式,结合归一化的梯度流方法,我们提出了杂交的自适应有限元方法,其中基函数的选取是基于网格剖分形式和解的渐近展开信息的。对于更简单的奇异摄动两点边值问题,可以严格证明我们的方法是一致收敛的。对于时间上有初始层现象的偏微分方程,本文研究了奇异摄动的KuramotoSivashinsky方程。该波动方程描述的是二维自由界面问题中火焰锋面的传播情况,在奇异摄动情形,该方程的解具有时间多尺度信息,具体来说,该方程的解除了会渐近收敛到Kuramoto-Sivashinsky方程的解外,还会有一个额外的初始层,这会导致解关于时间的二阶(或更高阶)导数会很大。我们分别提出了隐显Fourier谱方法和指数波动积分因子Fourier谱方法,并严格证明了它们的一致收敛性。特别地,后者在好初值情形时关于时间是一致且最优的二阶收敛的。
[Abstract]:Among the mathematical and physical problems in many fields of science and engineering, there are various boundary layer and inner layer phenomena, such as viscous boundary layer in incompressible fluid, internal shock layer in compressible gas dynamics. The initial layer of the hyperbolic equation, etc. An important feature of such problems is that in a very narrow region, The solution itself or its space / time derivative will change very quickly. When the traditional method of calculation is used to deal with this kind of problem, its computational cost will be extremely high. In this paper, the asymptotic analysis and other mathematical tools are applied to the numerical solution of partial differential equations with boundary layer / inner layer phenomenon. An efficient numerical method with uniform convergence is obtained. For the partial differential equations with boundary layer and inner layer phenomena in space, the Oseen flow problem with large Reynolds number and the radiation diffusion problem in multi-medium are studied in this paper. And the singularly perturbed nonlinear eigenvalue problem derived from Bose-Einstein condensates. The Oseen flow is a linear approximation of the incompressible Navier-Stokes flow. We transform the Oseen flow problem into an equivalent boundary value problem of the second order elliptic equation on a bounded computational domain by using the technique of equation decomposition and artificial boundary method. The numerical results show that our method is stable for Reynolds number. The phenomenon of boundary layer and inner layer can be captured in coarse grid. The nonequilibrium radiation diffusion equations describe the process of inertial confinement fusion. We propose a monotone tailor-made finite point method for numerical solution. The numerical results show that our method can capture the propagation of steep-front in rough mesh. The singularly perturbed nonlinear eigenvalue problem in this paper describes the Bose-Einstein condensation phenomenon under strong exclusion. By using the matching asymptotic expansion method, we give the asymptotic expansion form of the solution near the boundary layer and the inner layer. In combination with the normalized gradient flow method, we propose a hybrid adaptive finite element method. The selection of the basis function is based on the asymptotic expansion information of the mesh partition form and the solution. For a simpler singularly perturbed two-point boundary value problem, It can be proved strictly that our method is uniformly convergent. In this paper, the KuramotoSivashinsky equation of singularly perturbed is studied. The wave equation describes the propagation of flame front in the two-dimensional free interface problem. In the case of singularly perturbed, the solution of the equation has time and multi-scale information. The solution of the equation will converge asymptotically to the solution of the Kuramoto-Sivashinsky equation, and there will be an additional initial layer. This will result in a large second (or higher) derivative of the solution for time. We present the implicit Fourier spectral method and the exponential wave integral factor Fourier spectral method, respectively, and strictly prove their uniform convergence. In the case of good initial value, the latter is the second order convergent with respect to time consistency and optimization.
【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

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