Fourth‐order finite difference methods for three‐dimensional general linear elliptic problems with variable coefficients

作者：U. Ananthakrishnaiah, R. Manohar, John Ware Stephenson · 发表于：Numerical Methods for Partial Differential Equations · 年份：1987 · DOI：10.1002/num.1690030307 · 被引用次数：66 · 研究领域：Numerical methods for differential equations、Differential Equations and Numerical Methods、Advanced Numerical Methods in Computational Mathematics

Abstract In this article, we give a simple method for deriving finite difference schemes on a uniform cubic grid. We consider a general, three‐dimensional, second‐order, linear, elliptic partial differential equation with variable coefficients. We derive two simple fourth‐order schemes. When the coefficients of the second‐order mixed derivatives are equal to zero, the fourth‐order scheme requires only 19 grid points. When the coefficients of the mixed derivatives are not equal to zero and the coefficients of U xx , U yy , and U zz are equal, we require the 27 points of the cubic grid. Numerical examples are given to demonstrate the performance of the two schemes derived. There does not exist a fourth‐order scheme involving 27 grid points for the general case.