摄动有限差分方法研究进展

ADVANCES IN PERTURBATION FINITE DIFFERENCE (PFD) METHOD

振动有限差分（ＰＦＤ）方法，既离散徽商项也离散非微商项（包括微商系数），在微商用直接差分近似的前提下提高差分格式的精度和分辨率．ＰＦＤ方法包括局部线化微分方程的摄动精确数值解（ＰＥＮＳ）方法和摄动数值解（ＰＮＳ）方法以及考虑非线性近似的摄动高精度差分（ＰＨＤ）方法。论述了这些方法的基本思想、具体技巧、若干方程（对流扩散方程、对流扩散反应方程、双曲方程、抛物方程和ＫｄＶ方程）的ＰＥＮＳ、ＰＮＳ和ＰＨＤ格式，它们的性质及数值实验．并与有关的数值方法作了必要的比较．最后提出值得进一步研究的一些课题．

In the perturbation finite difference (PFD) method both the differentials and non- differential terms in the differential equation studied are discretized. High accuracy and high resolution difference schemes are obtained with the differentials being approximated by the direct differences (i.e., second-order-accurate center difference and first-order-accurate upwind difference). PFD method includes the perturbation exact numerical solution (PENS) scheme for locally lin- earized differential equation and the perturbation high-order-accurate difference (PHD) schemes with nonlinear effects. The basic idea and concrete technique of PFD method and some PENS-, PNS-and PHD- schemes for some differential equations including the convective-diffusion equation, convective- diffusion- react ion equation, hyperbolic equation, parabolic equation and K dV equation are summarized. The properties and typical numerical results of the above-mentioned schemes are discussed, comparisons of PFD method with the Other related numerical methods are made and some subjects for further studies are presented.