边界条件对对流扩散方程数值稳定性的影响

DISCUSSION ON THE CONVECTIVE NUMERICAL STABILITY OF CONVECTION-DIFFUSSION EQUATION-THE EFFECT OF THE BOUNDARY CONDITIONS

本文利用数值计算方法对采用均分网格的一维线性无源的对流－扩散方程在各种边界条件下的稳定性进行了分析，燕求出了不同边界条件下一维问题的中心差分和ＱＵＩＣＫ格式的临界网格Ｐｅｃｌｅｔ数。指出按现有方法得出的临界网格Ｐｅｃｌｅｔ数是判别差分格式对流数值稳定性的最苛刻的要求。对中心差分和ＱＵＩＣＫ格式，除两点边值问题以外的其它边界条件下的稳定性范围均不小于或远远大于两点边值问题的稳定性范围。通过计算还得出了格式的数值稳定性主要取决于计算区域下游侧的边界条件类型而与计算区域上游侧的边界条件类型无关的结论。

The stability of one-dimensional discretized convection-diffusion equation was analyzed at different boundary conditions. All the existing analysis methods are based on five assumptions: one-dimensional, linear, source-free, uniform grid and first kind boundary condition. It is found that the critical grid Peclet number based on the existing analysis method is the most severe requirement for convective stability Computations were carried out for CDS and QUICK schemes, and much larger critical grid Peclet number were obtained for non-first king boundary conditions. It is also found that the stability of the discretized scheme is only dependent on the boundary condition at the downstream end of the computational region.