对流扩散方程的数值流形格式及其稳定性分析

Numerical Manifold Scheme for Convection Diffusion Equation and Its Stability Analysis

将流形方法应用于对流扩散方程的数值求解,建立了基于标准Galerkin加权余量法的定常无源对流扩散方程的数值流形格式,采用一维定常无源对流扩散方程证明了物理覆盖的覆盖函数取完全一阶多项式的标准流形格式具有绝对的数值稳定性,并通过与一维对流扩散方程有限元解、精确解的对比,对该数值流形格式的稳定性进行了验证.同时,将基于四节点矩形有限单元覆盖系统的数值流形格式应用于二维平行管道中定常热对流扩散问题的数值分析.结果表明:在小的单元Pe(Pe<2)时,流形解的精度较有限元方法显著提高;在较大单元Pe条件下,一阶多项式覆盖函数的标准流形格式虽然绝对稳定,但假扩散作用显著,得到的数值解与真实结果存在较大的偏差.

The manifold method was employed to solve convection diffusion problems and the numerical manifold schemes for the convection diffusion equation were derived based on the Galerkin weighted residuals method.The standard manifold schemes with the first order polynomial function for physical cover were proved to be unconditionally stable,and the stability and adaptability of the present manifold schemes were confirmed by comparative analysis of numerical manifold solutions,finite element solutions and analytic solutions for one-dimensional steady source-free convection diffusion.The manifold schemes based on a four-node rectangular finite element cover system were used to simulate two-dimensional thermal convection-diffusion in pipe entry flow.The results show that the numerical manifold method can significantly improve computational accuracy at low element Peclet number（Pe〈2） compared with the finite element method.However,severe false diffusion effects at high element Peclet number will reduce computational accuracy and lead to erroneous results.