On the convergence and accuracy of homogeneous difference schemes for one-dimensional and multidimensional parabolic equations
On the convergence and accuracy of homogeneous difference schemes for one-dimensional and multidimensional parabolic equations
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一维和多维抛物方程齐次差分格式的收敛性和精度
DOI：
10.1016/0041-5553(63)90534-2
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发表时间：
1963
期刊：
影响因子：
--
通讯作者：
A. A. Samarskii
中科院分区：
文献类型：
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作者：
A. A. Samarskii
齐次差分格式（其一般定义在[I]中给出）是从其应用于具有一个空间变量的抛物型方程的角度来考虑的，[21-151.由于差分格式的收敛问题可以简化为线性方程解相对于其右侧、边界和初始数据的稳定性问题，因此在[21和[41]中首先获得了先验估计，由此得出稳定性。与[II]中一样，特别注意了差分方程右侧估计范数的选择，借助该范数可以在微分方程的不连续系数类中证明齐次格式的收敛性。在[31]中，[2l]中获得的先验估计被用于证明一致收敛以及估计具有不连续系数的热传导线性方程的齐次差分格式的精度阶。在[51]中，研究了具有III类边界条件的抛物型非线性方程（1）的齐次格式。在本文中，我们考虑具有一个或多个空间变量的拟线性抛物线方程的齐次格式。 5 1 中研究一维问题，0 2 中研究多维问题。9 1 中考虑导热系数 k= k (x, t, u) 的导热方程。本节改进了四点隐式格式（前向格式）和六点对称隐式格式的主要先验估计 [21 和 [41]。这使得
Homogeneous difference schemes, the general definition of which is given in [I], were considered from the point of view of their application to equations of the parabolic type with one space variable in [21-151. Since the problem of the convergence of difference schemes can be reduced to the problem of the stability of the solution of a linear equation with respect to its right hand side, and to the boundary and initial data, a priori estimates were obtained in the first instance in [21 and [41 from which the stability follows. As in [II, special attention was paid to the choice of norms for the estimation of the right hand side of the difference equation with the help of which the convergence of homogeneous schemes could be proved in the class of discontinuous coefficients of the differential equation. In [31 the a priori estimates obtained in [2l were used in the proof of the uniform convergence and in the estimate of the order of accuracy of homogeneous difference schemes for the linear equation of heat conduction with discontinuous coefficients. In [51 homogeneous schemes were studied for a non-linear equation (1) of the parabolic type with boundary conditions of kind III.In this paper we consider homogeneous schemes for quasilinear parabol ic equations with one or more space variables. One-dimensional problems are studied in 5 1 and multidimensional ones in 0 2. The equation of heat conduction with the coefficient of heat conduction k= k (x, t, u) is considered in 9 1. The main a priori estimates of [21 and [41 for a four-point implicit scheme (forward scheme) and for a six-point symmetrical implicit scheme are improved in this section. This makes it