We put forward different estimators of the local error
of split-step time integrators for evolution equations,
to act as a basis for adaptive time-stepping.
The first estimator is constructed from related pairs
of splitting formulae of different orders, similarly to
embedded Runge-Kutta methods, where several of the
compositions coincide to save computational work.
The second class of estimators is based on the defect
correction principle and yields asymptotically correct
estimates. The underlying idea is to form the defect
of the splitting approximation and backsolve for the
estimator using a related Sylvester equation.
This results in an integral representation which
can be approximated numerically with little computational
effort. We demonstrate that both error indicators can
successfully act as the basis for adaptive time-stepping
which is commensurate with the solution behavior for a
number of linear and nonlinear test problems comprising
nonlinear Schroedinger equations and dissipative parabolic