  We present strongly convergent explicit and semi-implicit adaptive numerical
schemes for systems of stiff stochastic differential equations (SDEs) where
both the drift and diffusion are non-globally Lipschitz continuous. This
stiffness may originate either from a linear operator in the drift, or from a
perturbation of the nonlinear structures under discretisation, or both. Typical
applications arise from the space discretisation of an SPDE, stochastic
volatility models in finance, or certain ecological models. We prove that a
timetepping strategy that adapts the stepsize based on the drift alone is
sufficient to control growth and to obtain strong convergence with polynomial
order. The order of strong convergence of our scheme is $(1-\varepsilon)/2$,
for $\varepsilon\in(0,1)$, where $\varepsilon$ becomes arbitrarily small as the
number of available finite moments for solutions of the SDE increases.
Numerically, we compare the adaptive semi-implicit method to a fully drift
implicit method, three tamed type methods and a truncated method. Our numerical
results show that the adaptive semi-implicit method is well suited as a general
purpose solver, is more robust than the explicit time stepping methods and more
efficient than the drift implicit method.
