  We introduce an explicit, adaptive time-stepping scheme for the simulation of
SPDEs with one-sided Lipschitz drift coefficients. Strong convergence rates are
proven for the full space-time discretisation with multiplicative trace-class
noise by considering the space and time discretisation separately. Adapting the
time-step size to ensure strong convergence is shown numerically to produce
more accurate solutions when compared to alternative fixed time-stepping
strategies for the same computational effort.
