求解热传导方程的一类新的加权差分格式

A Class of New Weighted Difference Schemes for Heat Conduction Equations

本文提出了数值求解热传导方程的一类新的Ｏ（ｋ，ｈ２）加权差分格式，并利用Ｆｏｕｒｉｅｒ方法讨论了格式的稳定性，证明了当１／（１＋ｅε）≤θ≤１时，格式是无条件稳定的，而当０≤θ＜１／（１＋ｅε）时，只有０＜ｒ≤ｆ（θ，ε），格式才稳定，其中ｆ（θ，ε）对任何固定的θ是正实数ε的严格单调增函数．最后通过数值算例检验了文中格式的高稳定性．

A class of new weighted difference schemes with higher stability is propose for solving one dimensional heat conduction equations in this paper.The method is based on the semi analytical method which discretizes the spatial derivative,having a truncation error of order O(1-2θ)k,k 2,h 2), where k and h represent the time and spatial steps respectively.The present method is unconditionally stable if 1/(1+e ε)≤θ≤1. However,when 0≤θ<1/(1+e ε) the method is stable only if 0<r≤f(θ,ε), where f(θ,ε) is an increasing function of its variable ε for any fixed θ ,which is weighting factors,and r=Dk/h 2. The results prove that our method is efficient.