稳定化谱元方法求解二维稳态对流扩散方程  

Stabilized spectral element method for solving a two-dimensional steady convection-diffusion equation

作　　者：和文强[1] 秦国良[1] 包振忠 王亚洲 HE Wenqiang;QIN Guoliang;BAO Zhenzhong;WANG Yazhou(Fluid Machinery Institute, Xi′an Jiaotong University, Xi'an 710049, China)

机构地区：[1]西安交通大学流体机械研究所,陕西西安710049

出　　处：《哈尔滨工程大学学报》2018年第3期446-451,共6页Journal of Harbin Engineering University

基　　金：国家自然科学基金项目(51776155)

摘　　要：为了求解对流项占优的对流扩散方程的非稳定性问题,本文提出了二维稳态对流扩散方程的一种高精度求解方法。该方法将一致逼近迎风方法和Chebyshev谱元方法相结合,形成了稳定化谱元方法。通过数值算例对该方法的可行性进行了验证,讨论了计算精度、收敛速度及边界层逼近和单元插值阶数的关系。研究表明:和谱元方法相比,稳定化谱元方法扩大了对流扩散方程的稳定求解区域;消除了计算区域内部及边界层附近的数值振荡,对内部和外部边界层都进行了很好的逼近;单元插值阶数的增加使计算精度及边界层的逼近效果都获得了迅速的提高。A numerical solution method with high accuracy is proposed herein for the instability problems of a two-dimensional steady convection-dominated convection-diffusion equation.The new scheme is a stabilized spectral element method based on the consistent approximate upwind method and the Chebyshev spectral element method.The feasibility of the method is verified by numerical examples.Moreover,the relation among the calculation accuracy,convergence rate,boundary layer approximation,and the order of the element interpolation is discussed.The results show that compared with the spectral element method,the stabilized spectral element method greatly enlarges the stable solution region for the convection-diffusion equation.The numerical oscillations in the computational domain and near the boundary layers in the convection-diffusion problem are completely eliminated.In addition,the internal and external boundary layers are excellently approximated.The computational accuracy and the effect of approximation to the boundary layers are both rapidly enhanced with the increase of the element interpolation order.

关 键 词：对流扩散方程 谱元法 稳定性 高精度 单元插值阶数 数值振荡 边界层 

分 类 号：O357.1[理学—流体力学]