  The main purpose of this paper is to give a solution to a long-standing
unsolved problem concerning the pathwise strong approximation of stochastic
differential equations with respect to the global error in the
$L_{\infty}$-norm. Typically, one has average supnorm error of order $\left(\ln
k/k\right)^{1/2}$ for standard approximations of SDEs with $k$ discretization
points, like piecewise interpolated Ito-Taylor schemes. On the other hand there
is a lower bound, which indicates that the order $1/\sqrt{k}$ is best possible
for spline approximation of SDEs with $k$ free knots. The present paper deals
with the question of how to get an implementable method, which achieves the
order $1/\sqrt{k}$. Up to now, papers with regard to this issue give only pure
existence results and so are inappropriate for practical use. In this paper we
introduce a nonlinear method for approximating a scalar SDE. The method
combines a Milstein scheme with a pieceweise linear interpolation of the
Brownian motion with free knots and is easy to implement. Moreover, we
establish sharp lower and upper error bounds with specified constants which
exhibit the influence of the coefficients of the equation.
