High-order difference schemes for two-dimensional elliptic equations

A high-order finite-difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equation. The approximation is defined on a square mesh stencil using nine node points and has a truncation error of order h⁴. Several test problems, including one modeling convection-dominated flows, are solved using this and existing methods. The results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.