Accuracy increase of finite difference calculations by means of differentiation of the partial differential equations and the boundary conditions

A numerical algorithm for producing high-order solutions for equilibrium problems is presented. The approximated solutions are improved by differentiating both the governing partial differential equations and their boundary conditions. The advantages of the proposed method over standard finite difference methods are: the possibility of using arbitrary meshes and an improvement in approximating the boundary conditions. Furthermore, the proposed method is capable of reaching approximate solutions which are more accurate than other finite difference methods, when the same number of nodal points participate in the local scheme. Calculations based on the proposed method and on another finite difference method are compared to analytical results. The comparison clearly illustrates the superiority of the proposed method.