求解对流扩散方程的紧致差分方法

A Compact Difference Scheme for Solving Convection Diffusion Equation

首先将指数变换u=pexpk2ε{x}以及降阶法和降维法相结合对常系数对流扩散方程构造了新的紧差分格式,给出了差分格式截断误差的表达式;并利用Fourier稳定性方法证明了该格式的稳定性,且收敛阶为O(τ2+h4).其次应用Richardson外推法对该紧差分格式外推一次得到O(τ4+h6)阶精度的近似解,最后通过数值算例说明该格式的有效性.

Firstly,a new compact difference scheme and its expression are constructed by means of the exponent transformation u = pexpk2ε{ x},p = veat,reduction of dimension and reduction of order. Secondly,the stability and the convergence order of the compact difference scheme are achieved by using Fourier method.Thirdly,Richardson＇s extrapolation method is successfully applied to the compact difference scheme and the approximate solution with accuracy O（ τ4＋ h6） is gained. Finally,the availability of the scheme is established by an example.