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

A numerical algorithm for increasing the accuracy of the approximated solutions of equilibrium problems is presented. 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 and the mesh is arbitrary. The basic idea of the proposed method is to reduce the required information from the surrounding grid nodes of the finite difference template by taking more information at the local level and applying differentiation on the governing partial differential equations and their boundary conditions. As a result the global characteristic matrix of the problem is considerably simplified. 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.