高维sine-Gordon方程高精度差分算法研究

发布时间：2018-01-07 17:28

本文关键词：高维sine-Gordon方程高精度差分算法研究　出处：《西南石油大学》2015年硕士论文　论文类型：学位论文

【摘要】：偏微分方程的数值计算是自然科学、技术科学和工程科学等重要学科和领域中不可或缺的工具,例如在气象学、石油勘探和航空航天技术等领域中,常常会涉及到各种各样的高维偏微分方程或方程组,因此求解这些偏微分方程已成为科学与工程计算的核心内容。高维sine-Gordon方程是一类被广泛应用于各个学科的非线性双曲型偏微分方程,随着科学技术和高性能计算机的发展和需要,研究高维sine-Gordon方程的无条件稳定、计算规模小而高精度的新型数值方法已成为当务之急。本文分别研究了二维和三维sine-Gordon方程的高精度有限差分方法。首先,将紧致差分格式与交替方向隐式格式相结合,得到方程的三层紧致交替方向隐式差分格式。其中紧致差分格式可以利用较少的计算节点达到较高的精度,而交替方向隐式格式可将高维问题转化为一系列一维问题,只需用追赶法求解系数矩阵为三对角矩阵的线性代数方程组。其次,通过能量分析法对紧致交替方向隐式差分格式的收敛性进行分析,证明了此格式具有二阶时间精度和四阶空间精度。最后,应用改进的Richardson外推算法,将格式的时间精度从二阶提高到四阶,并证明了外推后的差分格式具有四阶精度。数值算例结果表明紧致交替方向隐式差分格式和外推法改进后的格式分别达到了各自的精度。
[Abstract]:The partial differential equations of the numerical calculation is an indispensable tool for science, technology and Engineering Sciences and other important fields, such as in the field of meteorology, oil exploration and aerospace technology, often involves a variety of high dimensional partial differential equations or equations, thus solving the partial differential equation has become the core content of scientific and engineering computing. High dimensional nonlinear hyperbolic sine-Gordon equation is a kind of widely used in various disciplines of partial differential equations, with the development of science and technology and high performance computer development and the needs of unconditional stability of high dimensional sine-Gordon equations, the calculation model of small scale and high accuracy numerical methods has become a pressing matter of the moment.
High precision finite difference this paper studies the two-dimensional and three-dimensional sine-Gordon equation method. Firstly, the compact difference scheme and alternating direction implicit scheme combining the equations of three layers of compact alternating direction implicit difference scheme. The compact difference scheme can use less computing nodes to achieve high accuracy, and alternating direction implicit high-dimensional problem can be transformed into a series of one-dimensional problems, only after the linear algebraic equations method for solving the coefficient matrix of three diagonal matrix. Secondly, through energy analysis of compact alternating direction implicit convergence of difference schemes are analyzed, proved that this scheme has two order accuracy and four order spatial accuracy. Finally, the application of improved Richardson extrapolation method, the precision of time format from two to four order order, and it is proved that the extrapolated difference scheme with four order precision. The results of the numerical example show that the compact alternating direction implicit difference scheme and the improved form of the extrapolation method have reached their respective precision respectively.

【学位授予单位】：西南石油大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

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