Abstract

We propose an approach to construct monotonic finite difference schemes for solving the simplest elliptic and parabolic equations with the first derivatives and a small parameter at the highest derivative. For this, the notion of adaptive artificial viscosity is introduced. The adaptive artificial viscosity is used to construct monotonic difference schemes with the flow approximation of order \(O({{h}^{4}})\) for the boundary layer problem and \(O({{\tau }^{2}} + {{h}^{2}})\) for Burgers’ equation, where \(h\) and \(\tau \) are mesh steps in space and time, respectively. The Samarskii–Golant approximation (or upwind difference schemes) is used outside the region of high gradients. The importance of using schemes of second-order accuracy in time is outlined. The computational results are presented.

中文翻译：

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关于单调有限差分格式

摘要

我们提出一种构造单调有限差分方案的方法，以求解最简单的具有一阶导数和最高导数的小参数的椭圆和抛物线方程。为此，引入了自适应人工粘度的概念。自适应人工粘度用于构造单调差分方案，其边界层问题的流近似为\（O（{{h} ^ {4}}）\），阶数为\（O（{{\ tau} ^ { 2}} + {{h} ^ {2}}）\）用于Burgers方程，其中\（h \）和\（\ tau \）是分别在空间和时间上的网格步长。Samarskii–Golant近似（或迎风差分方案）用于高梯度区域之外。概述了使用及时的二阶精度方案的重要性。给出了计算结果。