粘弹性方程和Sobolev方程的一些数值研究

发布时间：2018-04-27 17:29

本文选题：粘弹性方程 + Sobolev方程　；参考：《南京航空航天大学》2017年硕士论文

【摘要】：本文基于有限差分方法对粘弹性方程和Sobolev方程建立差分格式。第一章,给出了本文研究背景和研究内容。第二章,首先对二维粘弹性方程建立了一个三层隐格式(S1)。为提高计算效率,在三层隐格式的基础上,提出了一个交替方向隐格式(S2)用于计算。用能量方法证明了格式S1和格式S2是唯一可解的,无条件稳定的,以平方模范数收敛,且收敛阶为O(τ~2+h~4)。最后数值实验结果验证了理论分析的正确性。第三章,将格式S1和格式S2推广到三维粘弹性方程,建立三维粘弹性方程的一个三层隐格式(S3)和一个交替方向隐格式(S4)。证明了格式是唯一可解的,无条件稳定的,以平方模范数收敛,且收敛阶为O(τ~2+h~4)。数值实验结果验证了理论分析的正确性。第四章,对Sobolev方程建立一个两层隐格式(S5),收敛阶为O(τ~2+h~4),证明了该格式是唯一可解的,以无穷模范数无条件收敛和稳定。数值实验结果验证了理论分析的正确性。
[Abstract]:In this paper, the finite difference scheme for viscoelastic equation and Sobolev equation is established based on finite difference method. In the first chapter, the research background and content are given. In the second chapter, a three-layer implicit scheme is established for the two-dimensional viscoelastic equation. In order to improve the computational efficiency, an alternating direction implicit scheme (S _ 2) is proposed on the basis of the three-layer implicit scheme. It is proved by the energy method that the schemes S1 and S2 are uniquely solvable, unconditionally stable, convergent with square model numbers, and the convergence order is O (蟿 ~ (2) H ~ (2) H ~ (4)). Finally, the results of numerical experiments verify the correctness of the theoretical analysis. In the third chapter, the scheme S1 and S2 are extended to the three-dimensional viscoelastic equation, and a three-layer implicit scheme (S3) and an alternating direction implicit scheme (S4) for the three-dimensional viscoelastic equation are established. It is proved that the scheme is uniquely solvable, unconditionally stable, convergent with square model numbers, and the convergence order is O (蟿 ~ (2) h ~ (4)). The results of numerical experiments verify the correctness of the theoretical analysis. In chapter 4, we establish a two-layer implicit scheme for Sobolev equation, the order of convergence is O (蟿 ~ (2) h ~ (4). It is proved that the scheme is solvable and unconditionally convergent and stable with infinite model number. The results of numerical experiments verify the correctness of the theoretical analysis.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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