非均匀网格的伸缩系数对紧致差分格式精度的影响分析

Analysis on the Influence of the Expansion Coefficient in the Non-Uniform Grids for the Accuracy of Compact Finite Difference Schemes

摘要: 在大梯度和边界层等问题数值计算中，均匀网格往往会导致计算量很大或计算误差较大，而非均匀网格在不增加计算量的同时，会将计算误差大大减小。非均匀网格上的非均匀程度可由伸缩系数进行控制，该系数不同，计算精度也不相同。本文针对一维对流扩散方程在非均匀网格上的紧致差分格式，分析了伸缩系数对计算结果的影响，并通过2个数值算例进行了验证。数值结果表明，伸缩系数存在最优值。合理选择最优伸缩系数，可以减小计算量，并能提高计算精度。另外，本文对紧致差格式在非均匀网格和均匀网格上的计算结果以及Crank-Nicolson格式进行了比较分析，表明非均匀网格上的紧致差分方法可以很好地解决大梯度和边界层问题的数值计算。

Abstract: In the numerical simulation of problems with large gradient and boundary layer, large amount of calculation or large calculation error will occur if uniform grids are used. Non-uniform grids can decrease calculation error greatly while keep the same computational expense. The non- uniformity of the non-uniform grids can be controlled by an expansion coefficient, which affects the accuracy of a scheme. In this paper, analysis has been presented for the effect of the expan-sion coefficient to the numerical results of one dimensional convective-diffusion equation by using compact difference scheme under non-uniform grids. Two numerical examples are given and it is indicated that optimal value of the expansion coefficient may exist. The computation accuracy can be increased greatly by choosing reasonable expansion coefficient. Moreover, comparisons among compact difference scheme on both non-uniform grids and uniform grids and Crank-Nicolson schemes show that the compact difference on non-uniform grids can be used to solve the problem of large gradient and boundary layer with high accuracy.