Technical note: A fourth-order finite difference scheme for a system of a 2D nonlinear elliptic partial differential equations

Abstract

In this article we present a fourth-order finite difference scheme, for a system of two-dimensional, second-order, nonlinear elliptic partial differential equations with mixed spatial derivative terms, using 13-point stencils with a uniform mesh size h on a square region R subject to Dirichlet boundary conditions. The scheme of order h4 is derived using the local solution of the system on a single stencil. The resulting system of algebraic equations can be solved by iterative methods. The difference scheme can be easily modified to obtain formulae for grid points near the boundary. Computational results are given to demonstrate the performance of the scheme on some problems including Navier-Stokes equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 43–53, 2001