Order h4 difference methods for a class of singular two space elliptic boundary value problems

In this article, we derive difference methods of O(h4) for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points. We obtain two sets of fourth-order difference methods; one in the absence of mixed derivatives, second when the coefficients of uxy are not equal to zero and the coefficients of uxx and uyy are equal. There do not exist fourth-order schemes involving nine grid points for the general case. The method having two variables has been tested on two-dimensional viscous, incompressible steady-state Navier-Stokes' model equations in polar coordinates. The proposed difference method for scalar equation is also applied to the Poisson's equation in polar coordinates. Some numerical examples are provided to illustrate the fourth-order convergence of the proposed methods.