On the one hand the explicit Euler scheme fails to converge strongly to the
exact solution of a stochastic differential equation (SDE) with a superlinearly
growing and globally one-sided Lipschitz continuous drift coefficient. On the
other hand the implicit Euler scheme is known to converge strongly to the exact
solution of such an SDE. Implementations of the implicit Euler scheme, however,
require additional computational effort. In this article we therefore propose
an explicit and easily implementable numerical method for such an SDE and show
that this method converges strongly with the standard order one half to the
exact solution of the SDE. Simulations reveal that this explicit strongly
convergent numerical scheme is considerably faster than the implicit Euler
scheme.