Fourth-order finite difference scheme for a system of quasilinear elliptic equations

The system of two quasilinear elliptic equations is approximated by the method of lines, which has the truncation error O(h²) at points neighboring the boundary and O(h⁴) at the most interior points. It is proved that the global error of the method is O(h⁴) at all mesh points. The two-point boundary value problem for the system of ordinary differential equations that arises from the method of lines is solved by the O(h⁴) convergent finite difference scheme, suitable to the equations of the form u_xx = f(x, u) without the first derivative u_x. The system of algebraic equations obtained by the full discretization is solved by Gauss elimination method for three diagonal matrices combined with the method of iterations. A numerical example is presented.