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 h 4 . 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.