Abstract

In this article, we present a study on constructing a second order local approximation monotone difference schemes on spatial non-uniform grids for the parabolic equation of convection-diffusion type with a third kind boundary condition without using the basic differential equation at the boundary of the domain. The goal is a combination of the differential inequality, the regularization principle and the assumption of the existence and uniqueness of a smooth solution. In this case, the boundary conditions are directly approximated with the second order on a two-point stencil. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved. With the help of difference maximum principle, two-sided estimates of the difference solution are established and an important a priori estimate in a uniform \(C\)-norm is obtained.

中文翻译：

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具有第三类边界条件的多维对流扩散问题的非均匀网格上的二阶单调有限差分格式

摘要

在本文中，我们提出了在空间非均匀网格上构造具有第三类边界条件的对流扩散型抛物线方程的二阶局部近似单调差分格式的研究，而无需在边界处使用基本微分方程。领域。目标是将微分不等式、正则化原理和光滑解的存在唯一性假设结合起来。在这种情况下，边界条件直接用两点模板上的二阶近似。证明了所提算法对原始二阶微分问题解的收敛性。借助差分最大值原理，\(C\) -范数得到。