【作者】 李建军；

【导师】 冯民富；

【作者基本信息】 四川大学 ， 计算数学， 2007， 硕士

【摘要】 一直以来,对流占优由于其重要的物理背景而成为一个研究热点。由于这类问题具有双曲性质,传统的差分法和有限元法经常出现伪数值振荡。二十世纪七十年代以后,诸多非标准有限元方法被相继提出。这些方法结构虽不同,但都有一个共同的着眼点:设法处理好对流项使数值算法在某种程度上能反映原问题对流占优的特性,避免伪数值振荡。其中比较重要的两种方法是有限体积法和子网格涡旋粘性法。有限体积法能够保持原物理问题的特性,特别是守恒性,并且其计算格式相对也较为简单,因此被广泛应用于数值求解流体动力学方程。然而,数值算例表明,对某些对流占优的扩散问题,用此法得出的数值解仍出现某种非物理震荡,稳定性不够理想;而基于粘性发展的子网格涡旋粘性法得到的解的稳定性很强,解的精度也能得到保证。为了利用eddy viscosity法和有限体积法的优点。本文试着将eddy viscosity法推广到FV格式,两者结合来解对流扩散方程。并对所导出的格式进行了理论分析。结果表明,文章所建立的格式的解的稳定性得到增强,有效地克服了FV法伪数值振荡现象,同时解的精度也达到了最优。更多还原

【Abstract】 The problem of convection dominated diffusion has been the research hotspot at present. Due to the hyperbolic character of this problem and the effect of the term of diffusion, conventional solution of Garlerkin finite element often appears false numeric surge. After 1970s, a good many non-standard finite element methods are discu- ssed. Although they have different configurations, they have a comm- on starting point:trying to deal with the term of convection, numeric computation to some extent can image the character of convection dominated and avoid numeric surge. There are comparatively impor- tant methods: finite volume method and subgrid scale eddy viscosity method. finite volume method can keep the character of original physical problem, especially for conservation, and also its calculation scheme is relatively easy; the stability of the solution which is get by subgrid scale eddy viscosity method is higher and the accuracy is improved. In order to utilize the advantages of finite volume method and subgrid scale eddy viscosity method and overcome some lack of each method, a kind of method which is combined finite volume method and subgrid scale eddy viscosity method to deal with the linear time-dependent convection dominated convection diffusion problem is proposed. It is showed by the theoretical analysis on this format that the method which is proposed in this paper can enhance the stability of the solution and overcome the phenomenon of false numeric surge. Meanwhile, the accuracy of the solution by subgrid scale eddy visco- sity method is gained.更多还原