Technical note: The numerical solution of the system of 3‐D nonlinear elliptic equations with mixed derivatives and variable coefficients using fourth‐order difference methods

Abstract

In this article, we report two fourth‐order difference methods for the numerical integration of the system of general 3‐D nonlinear elliptic equations subject to Dirichlet boundary conditions on a uniform cubic grid. When the coefficients of uxy, uyz, and uzx are not equal to zero and the coefficients of uxx, uyy, and uzz are equal, we require 27 grid points; when the coefficients of uxy, uyz, and uzx are equal to zero, we require only 19 grid points. The utility of the new methods is shown by testing the methods on various examples, including 3‐D steady state viscous incompressible Navier–Stokes' model equations and Poisson's equation in polar coordinates, which confirm the accurate and oscillation‐free solutions for large Reynolds numbers even in the vicinity of singularity. © 1995 John Wiley & Sons, Inc. Copyright © 1995 Wiley Periodicals, Inc.