Fourth order nine point unequal mesh discretization for the solution of 2D nonlinear elliptic partial differential equations

We propose a nine point fourth order accurate compact difference scheme with unequal mesh size in different coordinate directions and discuss line iterative methods for the solution of elliptic partial differential equations with variable coefficients subject to appropriate Dirichlet boundary conditions. We also prove the convergence of line iterative methods for solving the linear system arising from proposed discretization of a two dimensional diffusion-convection equation. The proposed method is successfully applied to solve singular elliptic equation, model Burgers' equation and Navier Stokes equations of motion in a rectangular domain. Finally, we perform numerical experiments to demonstrate the high accuracy and stability advantages of the proposed new scheme.