混合解析/数值方法在含高频振荡点源项的欧拉方程数值计算上的应用

 通过模型方程分析看到了一个重要的现象, 如果源项涉及的时间尺度远小于对流项的时间尺度, 那么基于对流时间尺度作为步长的传统数值方法, 即使源项相对于墩流项是个小量, 也会导致平均尺度上错误的结果。为了克服这种困难, 采用时间分裂方法, 把方程分裂成含对流项部分的偏微分方程( PDE) 和包含源项的常微分方程(ODE) 基础上, PDE 使用传统的数值方法, ODE 用解析的方法求解。该混合方法在数值格式时间步长小于平均流动时间尺度时, 得到正确解, 而与点源所隐含的时间尺度无关。把这个方法应用在含源流动的Euler 方程的计算中, 计算了翼型振荡问题, 取得了理想的结果。

Mixed Analytical/Numerical Method Applied to Euler Equations with High Frequence Oscillating Source Terms

A model equation shows that the conventional method gives very wrong results for the time step just based on the advection part, if the time scale of the source term is much smaller than that of the advection part. Even if the source term i s a small quantity compared with the advection part, the large scale structure is completely wrong. In order to overcome the numerical difficulty, consider the operator splitting method. After splitting, the mean flow part (advection part) will be treated using any suitable numerical method, while the source step (ODE ) is solved analytically. With such a method, one can use a time step just base d on the time scale of the advection part to capture the correct large scale, in dependently of how small the time scale of the source term is. This method is ap plied to the Euler equations with an oscillation source term. The satisfying res ults are given.