基于一类非标准基底的结构化的谱元方法及其应用

发布时间：2018-03-04 14:47

本文选题：高阶问题　切入点：新的谱和谱元方法　出处：《江苏师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：有限差分法、有限元方法和谱方法是求解偏微分方程的三种主要方法.谱方法是以整体光滑的正交多项式作为基底逼近问题的解,其优点是高精度,也就是说,只要问题的解越光滑,逼近的程度越好.在本文中,我们针对高阶问题提出一种新的谱和谱元方法.首先,对于带齐次边界条件的常系数的四阶问题,我们利用矩阵正交分解的技巧构造了一类新的基函数.其次,基于新的基函数,我们提出一类新的时空谱方法,所提算法在保证高精度的同时节省了计算时间.我们证明了该方法在空间方向和时间方向上同时具有高精度,数值结果也验证了理论分析.最后,我们利用新的基函数的优点,对于一维和二维四阶问题提出了新的谱元方法的求解格式,数值结果揭示了谱元方法的有效性.
[Abstract]:The finite difference method, finite element method and spectral method are three main methods for solving partial differential equations. As long as the solution of the problem is smoother, the degree of approximation is better. In this paper, we propose a new spectral and spectral element method for higher order problems. We construct a new class of basis functions by using the technique of matrix orthogonal decomposition. Secondly, based on the new basis functions, we propose a new kind of space-time spectral method. It is proved that the proposed method has high accuracy in both spatial and temporal directions, and the numerical results verify the theoretical analysis. Finally, we use the advantages of the new basis function. A new spectral element method is proposed for solving one-dimensional and two-dimensional fourth-order problems. The numerical results show the validity of the spectral element method.
【学位授予单位】：江苏师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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