NS方程激波计算的摄动有限差分方法

摄动有限差分(PFD)方法从一阶迎风差分格式出发,将差分系数展开为网格步长的幂级数,通过提高修正微分方程的逼近精度来获得更高精度的差分格式。由于格式基于一阶迎风格式,因此具有迎风效应、网格节点少等特点。本文首先通过对Burgers方程的摄动差分格式的推导,将摄动有限差分格式引入时间相关法的计算,并构造了守恒形式的摄动有限差分格式,然后推广到一维Navier-Stokes方程组的计算。数值比较研究表明:本文构造的NS方程摄动有限差分格式具有比一阶迎风较高的精度和分辨率,而且保持了一阶迎风格式的无振荡性质。

Peturbational finite difference scheme for shock-wave computing of Navier-Stokes equations

In the peturbational finite difference(PFD) method,the difference coefficients of the first-order accurate upwind difference scheme are expanded into the power series of grid size,by improving the approach accuracy of modified differential equation to obtain higher-order accurate difference scheme.PFD scheme has upwind effect and only uses three grids as in the first-order upwind difference scheme.In this paper,the PFD scheme of the Burgers equation is derived.Then combined with the time depending method,the conservative-type PFD scheme is constructed and generalized to compute one-dimensional Navier-Stokes equations.The numerical results show that the present PFD scheme of NS equations has higher order accurate and better resolution than the first-order accurate upwind scheme does,and can remain the essentially non-oscillatory property.