  A new explicit stochastic scheme of order 1 is proposed for solving
commutative stochastic differential equations (SDEs) with non-globally
Lipschitz continuous coefficients. The proposed method is a semi-tamed version
of Milstein scheme to solve SDEs with the drift coefficient consisting of
non-Lipschitz continuous term and globally Lipschitz continuous term. It is
easily implementable and achieves higher strong convergence order. A stability
criterion for this method is derived, which shows that the stability condition
of the numerical methods and that of the solved equations keep uniform.
Compared with some widely used numerical schemes, the proposed method has
better performance in inheriting the mean square stability of the exact
solution of SDEs. Numerical experiments are given to illustrate the obtained
convergence and stability properties.
