  For a stochastic differential equation driven by a fractional Brownian motion
with Hurst parameter $H> \frac12$ it is known that the classical Euler scheme
has the rate of convergence $2H-1$. In this paper we introduce a new numerical
scheme which is closer to the classical Euler scheme for diffusion processes,
in the sense that it has the rate of convergence $2H-\frac12$. In particular,
the rate of convergence becomes $\frac 12$ when $H$ is formally set to $\frac
12$ (the rate of Euler scheme for classical Brownian motion). The rate of weak
convergence is also deduced for this scheme. The main tools are fractional
calculus and Malliavin calculus. We also apply our approach to the classical
Euler scheme.
