  Time-fractional parabolic equations with a Caputo time derivative are
considered. For such equations, we explore and further develop the new
methodology of the a-posteriori error estimation and adaptive time stepping
proposed in [7]. We improve the earlier time stepping algorithm based on this
theory, and specifically address its stable and efficient implementation in the
context of high-order methods. The considered methods include an L1-2 method
and continuous collocation methods of arbitrary order, for which adaptive
temporal meshes are shown to yield optimal convergence rates in the presence of
solution singularities.
