关于热传导方程半离散差分格式的一个注记

本文研究了热传导方程初边值问题的半离散化差分格式直接解算法.分别从Dirichlet和Neumann边界条件出发,直接由空间差分格式导出与时间相关的一阶常微分方程组,随后通过正/余弦变换获得了原方程的半解析解,并给出了相关收敛性分析.并对中心差分格式和紧差分格式的精度差异,通过矩阵特征值理论给出了相关原因分析.另外,对于二维热传导方程初边值问题,应用矩阵张量积运算,该直接解算法可直接演变成二重正(余)弦变换.该方法由于不涉及时间上的离散,从而具有较好的计算效率.

A note on semi-discrete difference schemes of heat conduction equations

In this paper, based on semi-discrete difference scheme, a new direct algorithm on heat conduction equations is presented. For the Dirichlet and Neumann boundary conditions, the ordinary differential equations are derived from spatial difference scheme, and then the semi-analytical solution of original equation is obtained by the sine or cosine transforms. The corresponding convergence is also analyzed. The precision analysis on central difference and compact difference schemes is also presented by the corresponding eigenvalue theory. In addition, this direct algorithm may be directly developed into double sine or cosine transforms for two-dimensional heat conduction equations by the matrix tensor product. Since there does not exist the time discrete such that it has very high computational efficiency.