Single Cell Discretization of O(h4) for the Numerical Solution of the System of Two-dimensional Non-linear Elliptic Boundary Value
by
R.K. Mohanty
Department of Mathematics, University of Delhi, Delhi - 110 007, INDIA

We report new finite difference methods of O(h⁴) accuracy for the numerical solution of two dimensional non-linear elliptic equation A(x, y)U_xx+2B(x, y)U_xy+C(x, y)U_yy = f(x, y, U, U_x, U_y) on a uniform square grid and the estimates of u_n at each internal grid points subject to the Dirichlet boundary conditions. The proposed methods require only nine grid points ( single-cell form ) and applicable to the problems both in Cartesian and polar coordinates. We also discuss two sets of fourth order difference methods ( case-I : when B=0 and case-II : when A=C, B =/= 0 ). There do not exist fourth order finite difference methods for the general case using nine grid points ( single-cell form ). We establish, under appropriate conditions the fourth order convergence of the proposed method. We also extend our technique to the system of non-linear elliptic equations with variable coefficients. The proposed methods have been tested on two-dimensional viscous, incompressible steady-state Navier-Stokes’ model equations both in Cartesian and polar coordinates. The proposed difference method for scalar equation is also applied to the Poisson’s equation in polar coordinates. Numerical results are provided to illustrate the fourth order convergence of the proposed methods.

Date received: October 31, 2002