对流扩散方程的拟谱格式稳定性分析及扩稳方法

针对拟谱方法在高雷诺数下的格式非稳定性问题,通过特征分析及算例验证系统研究对流扩散方程的拟谱方法稳定性,重点采用罚方法处理边界条件来拓宽稳定性范围.研究表明:拟谱方法在处理强约束边界条件时稳定性条件苛刻,是无法计算高雷诺数问题的根本原因;罚方法处理后的边界条件问题可以将拟谱方法的稳定范围扩宽1～3个数量级,极大地提高了拟谱方法所能求解的雷诺数上限;此外,罚方法还能保持拟谱方法的高精度优点,对减少计算时间亦有裨益.研究工作对流动传热传质方程的拟谱格式的数学特性分析及设计具有理论指导意义.

Stability Analysis and Stability Widening Method for Pseudospectral Schemes of Advection-Diffusion Equation

The schemes of the pseudospectral method are instable at high Reynolds numbers.In this study,the scheme stability of the advection-diffusion equation was investigated through the eigenvalue analysis theory and the numerical example verification.The emphasis was laid on using the penalty method to handle the boundary conditions for the purpose of widening the stability range of the pseudospectral schemes at high Reynolds numbers.The results show that the stability conditions under strongly restrained boundary conditions are so hard that the schemes will fail at high Reynolds numbers.However,when the penalty method is adopted to handle the boundary conditions,the stability range can increase by 1-3 orders of magnitude and the upper limit of Reynolds numbers can rise significantly.In addition,the penalty method can keep the high accuracy property of the pseudospectral method and is helpful in reducing the computational cost.The present study may be of theoretical importance for the mathematical analysis and design of the pseudospectral schemes in the numerical simulation of heat and mass transfer and fluid flow governing equations.