High‐order methods for elliptic equations with variable coefficients
High‐order methods for elliptic equations with variable coefficients
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DOI：
10.1002/num.1690030306
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发表时间：
1987-09
影响因子：
3.9
通讯作者：
U. Ananthakrishnaiah;R. Manohar;J. W. Stephenson
中科院分区：
文献类型：
--
作者：
U. Ananthakrishnaiah;R. Manohar;J. W. Stephenson
在本文中，我们给出了一种在均匀方形网格上开发有限差分格式的简单方法。我们考虑一个具有可变系数的一般二维二阶偏微分方程。在九点方案的情况下，我们以相当优雅的方式获得了 Young 和 Dauwalder 的已知结果。我们展示了如何扩展它以获得十三个点的四阶方案。我们推导了两个这样的方案，它们很有吸引力，因为它们可以很容易地适应以获得边界附近的网格点的公式。除此之外，这些公式仅需要对典型的强迫函数进行九次评估。给出了数值例子来证明四阶方案之一的性能。
In this article, we give a simple method for developing finite difference schemes on a uniform square gird. We consider a general, two-dimensional, second-order, partial differential equation with variable coefficients. In the case of a nine-point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth-order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily bnto obtain formulas for gird points near the boundary. In addition to this, these formulas only require nine evaluations for the typical forcing function. Numerical examples are given to demonstrate the performance of one of the fourth-order schemes.