A new finite difference discretization of order four for (∂u/∂n) for two-dimensional quasi-linear elliptic boundary value problem

We report a new algorithm of O(h4) for the numerical solution of ux and uy for the solution of two dimensional quasi-linear eiliptic equation A(x,y,u)uxx + 2B(x,y,u)uxy + C(x,y,u)uyy = f(x, y,u,ux,uy) subject to the Dirichlet boundary conditions. The proposed method requires only nine grid points on a uniform square grid and applicable to the problems both in cartesian and polar coordinates. We also discuss two sets of fourth-order finite difference methods; one in the absence of mixed derivative term, second when the coefficient of uxy is not equal to zero and the coefficients of uxx and uyy are equal. There do not exist fourth order finite difference schemes involving nine grid points for the general case. Numerical examples are given to illustrate the method and its fourth order convergence.