  We introduce a new approach for designing numerical schemes for stochastic
differential equations (SDEs). The approach, which we have called direction and
norm decomposition method, proposes to approximate the required solution $X_t$
by integrating the system of coupled SDEs that describes the evolution of the
norm of $X_t$ and its projection on the unit sphere. This allows us to develop
an explicit scheme for stiff SDEs with multiplicative noise that shows a solid
performance in various numerical experiments. Under general conditions, the new
integrator preserves the almost sure stability of the solutions for any
step-size, as well as the property of being distant from $0$. The scheme also
has linear rate of weak convergence for a general class of SDEs with locally
Lipschitz coefficients,and one-half strong order of convergence.
