Prandtl's boundary value problem on a flat plate for a system of boundary layer equations is quasilinear. The solution of such a problem has complicated behaviour. Experience in constructing satisfactory numerical methods for problems with a parabolic boundary layer allows us to develop a ``natural'' special finite difference scheme for the Prandtl problem; if it is assumed that the coefficients multiplying the derivatives in the transport equation for Prandtl's problem are known, then we have a linear transport equation, and under appropriate conditions on these coefficients, the scheme for Prandtl's problem converges $\eps$-uniformly, where $\eps=Re^{-1}$, and $Re$ is the Reynolds number. It should be noted that at present $\eps$-uniformly convergent difference schemes for Prandtl's problem are unknown. Nevertheless, it is of great interest to study numerically the above-mentioned special scheme. Numerical experiments indicate $\eps$-uniform convergence, with respect to both the number of grid nodes and the number of iterations required for convergence of the iterative process, of both the numerical solution and its discrete derivatives, outside a neighbourhood of the leading edge of the plate.