无源对流扩散方程的两类修正差分格式

本文研究含参数ε的无源对流扩散问题的有限差分格式.首先在三点模板上将两边结点处的函数值关于中心点进行泰勒展开,反复利用原微分方程,通过"降阶"的思想将两个泰勒展式中的高阶导数项化为只含一阶导数的展式,联立展式消去一阶导数项从而得到形式上精确的差分格式.由于形式上精确的差分格式的系数含无穷项,如何保留有限项使得差分格式分别适用于求解参数较大或参数较小的对流扩散问题是本文研究的重点,为此本文分情形设计了两类差分格式:当参数较大时,因h的幂次对差分格式系数影响更大,本文设计出"横向系列修正差分格式(HDS)",其精度分别可达到二阶、四阶、六阶、八阶;而对小参数问题,相对于步长, 1/ε的幂次对差分格式的系数影响更大,据此本文设计出"纵向系列修正差分格式(VDS)".数值算例将横向、纵向系列格式与七种参考文献给出的差分格式进行了数值比对,验证了本文设计的横向差分格式(HDS)适用于求解ε较大时的对流扩散问题,而纵向系列修正差分格式(VDS)适用于求解ε较小时的问题,且数值解精度较参考格式更高.

Two Kinds of Modified Difference Schemes for ConvectiveDiffusion Equations Without Source Term

In this paper, equidistance difference schemes for the convection diffusion equation without source term are studied. The difference schemes are designed on the three-point template. After expanding the function values at both nodes about the center point by Taylor’s expansion, two Taylor expansions are obtained. While the original differential equation is used repeatedly, the higher derivative terms in two Taylor expansions are transformed into expansions containing only the first-order derivative term by means of the idea of "reduced order". Then the first-order derivative can be eliminated combining the two Taylor expansions and a formally accurate difference scheme can be obtained. Since the coefficients of the difference scheme are composed of infinite series, how to preserve finite terms to make the difference scheme suitable for problems with large or small parameters is the focus of this paper.We design two kinds of difference schemes in different situations: when the parameter is large, the power of h has a greater impact on the difference scheme coefficient, so we design a kind of "horizontal series modified difference schemes(HDS)", whose accuracy can reach the second order, the fourth order, the sixth order, the eighth order respectively. However, when the parameter ε is very small, the power of 1/ε has a greater impact on the difference scheme coefficient than the step size, therefore we design a kind of "vertical series modified difference schemes(VDS)". One numerical example is selected to carry on the experiment, and the numerical comparisons are made among the HDS, VDS and the seven difference schemes given in the references. Results show that the horizontal difference schemes(HDS) designed in this paper are suitable for the case where ε is larger, and the vertical series modified difference schemes(VDS) are suitable for the case where ε is very small. And it is also showed that the accuracy of our method is better than that of the difference schemes in references.