摄动有限差分(PFD)方法的数值计算

Numerical simulation of perturbational finite difference (PFD) method

本文利用数值摄动的思想发展了对流扩散(CD)方程的摄动有限差分(PFD)[2]格式,局部线化可获得摄动准确解(PENS)格式。把格式中的指数函数按网格雷诺数的幂级数展开,能够给出PENS格式的各阶近似,即各阶PFD格式。本文构造了一维CD方程的PENS及二阶PFD格式以及二维CD方程的二阶PFD格式,并利用二阶精度的PFD格式计算了一维CD方程、方腔流动和前、后台阶绕流流动等算例。计算给出了准确度较高的数值结果。

The perturbational finite difference (PFD) method is a kind of high accurate compact difference method, but its idea is different from the normal compact method. This method can get a perturbational exact numeical solution (PENS) scheme for locally linearized convectivediffusion (CD) equation. PENS scheme (or method) is similar to the finite analytical (FA) method and exact difference scheme (EDS), they are all exponential schemes, but PENS scheme is much simpler. By expanding the exponential term in PENS to the powerseries of grid Reynolds number, the approximate schemes of different order of PENS scheme, i.e. PFD schemes, can be obtained. They are all upwind schemes and remain the concise structure form of firstorder upwind scheme. The PENS and PFD schemes for 1D CD equation are constructed, and PFD scheme is extended to 2D CD equation. For 1D equation and 2D flowing problems, highly accurate results are obtained by using second order accurate PFD schemes.