非定常对流扩散方程的有理型高精度紧致差分方法

发布时间：2018-03-29 18:40

本文选题：非定常对流扩散方程　切入点：有理型高精度紧致格式　出处：《宁夏大学》2015年硕士论文

【摘要】：本文主要建立了求解对流扩散方程的有理型高精度紧致(RHOC)差分方法.首先在空间上,基于函数的泰勒级数展开和空间四阶紧致差分公式,推导了一维定常对流扩散方程的RHOC差分格式.然后,在时间上利用Crank-Nicolson格式进行离散,得到了求解一维非定常对流扩散方程的RHOC差分格式,该差分格式时间上具有二阶精度,空间上具有四阶精度.通过vonNeumann分析方法证明了RHOC格式是无条件稳定的.通过与其它几种已有格式的数值算例结果比较,验证了RHOC格式的精确性和稳定性.接着,基于一维问题的研究,分别推导出了二维和三维非定常对流扩散方程的交替方向隐式的有理型高精度紧致(RHOC ADI)差分格式,该差分格式时间上具有二阶精度,空间上具有四阶精度,并且是无条件稳定的.数值实验结果表明,本文针对非定常对流扩散方程所建立的RHOC ADI差分方法,不仅能够适用于非定常对流扩散问题,而且能够很好地求解非定常纯对流问题或纯扩散问题,并且其计算效果均优于其它的差分格式.该方法很好地结合了高精度紧致差分格式和ADI方法的优势,为求解非定常对流扩散方程提供了一类精确、稳定、高效的数值方法.最后,推导了一维定常对流扩散反应方程的RHOC差分格式,并利用Richardson外推法和算子插值技术将格式的精度提高到六阶.并通过数值实验验证了格式的精确性和可靠性.
[Abstract]:In this paper, the rational high precision compact RHOC difference method for solving convection-diffusion equations is established. Firstly, the Taylor series expansion based on function and the fourth order compact difference formula in space are established. The RHOC difference scheme for one dimensional steady convection-diffusion equation is derived, and then the RHOC difference scheme for solving one dimensional unsteady convection-diffusion equation is obtained by using Crank-Nicolson scheme in time. The difference scheme has second order accuracy in time. It is proved that the RHOC scheme is unconditionally stable by the vonNeumann analysis method. The accuracy and stability of the RHOC scheme are verified by comparison with the numerical examples of other schemes. Based on the study of one-dimensional problems, the alternating direction implicit rational compact RHOC ADI difference schemes for two-dimensional and three-dimensional unsteady convection-diffusion equations are derived respectively. The scheme has second order accuracy in time. The numerical results show that the RHOC ADI difference method for unsteady convection-diffusion equations can not only be applied to unsteady convection-diffusion problems. Moreover, the unsteady pure convection problem or pure diffusion problem can be solved well, and its calculation results are superior to those of other difference schemes. This method combines the advantages of the high precision compact difference scheme and the ADI method. A kind of accurate, stable and efficient numerical method is provided for solving unsteady convection-diffusion equation. Finally, the RHOC difference scheme of one-dimensional steady convection-diffusion reaction equation is derived. The Richardson extrapolation method and operator interpolation technique are used to improve the accuracy of the scheme to the sixth order, and the accuracy and reliability of the scheme are verified by numerical experiments.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

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