关于抛物型方程的一种线条法

On the Method of Lines for Parabolic Equations

本文对于热传方程运用线条法形成常微分方程系统:就空间变量的离散化,二阶偏导数项在邻接左、右边界的两根线条上采用三点对称格式〔截断误差为O(h^2)〕,在中间其余线条上采用五点对称格式〔截断误差为O(h^4)〕,注意到常微分方程组的系数阵具有全为负的本征值,本文应用A_0一稳定的P阶线性方法,证明数值解与偏微分方程精确解之间整体误差可达O(h^(4p)/(p+1)),接近O(h^4)。这表明,在有局部的低阶离散化误差的情况下仍可获得高阶的整体误差。

The heat-conduction equation is,semi-discreted by the following method of lines: the second order partial derivative term is approximated by three point symmetric scheme on two lines adjacent to boundaries (the discretization errors are 0(h2)). by five point symmetric scheme on (he intermediate lines (0(h4)). According as the coefficient matrix of ordinary differential equation system has eigenvalues all negative, it is able to make use of any pth order A,,-siable method to prove that the global error of numer ical solution reaches 0( ). approaching 0(h4). It shows thai the method of lines can get the higher global errors in the case of local lower discretization errors.