  Adaptive time stepping methods for metastable dynamics of the Allen Cahn and
Cahn Hilliard equations are investigated in the spatially continuous,
semi-discrete setting. We analyse the performance of a number of first and
second order methods, formally predicting step sizes required to satisfy
specified local truncation error $\sigma$ in the limit of small order parameter
$\epsilon \rightarrow 0$ during meta-stable dynamics. The formal predictions
are made under stability assumptions that include the preservation of the
asymptotic structure of the diffuse interface, a concept we call profile
fidelity. In this setting, definite statements about the relative behaviour of
time stepping methods can be made. Some methods, including all so-called energy
stable methods but also some fully implicit methods, require asymptotically
more time steps than others.The formal analysis is confirmed in computational
studies. We observe that some provably energy stable methods popular in the
literature perform worse than some more standard schemes. We show further that
when Backward Euler is applied to meta-stable Allen Cahn dynamics, the energy
decay and profile fidelity properties for these discretizations are preserved
for much larger time steps than previous analysis would suggest. The results
are established asymptotically for general interfaces, with a rigorous proof
for radial interfaces. It is shown analytically and computationally that for
most reaction terms, Eyre type time stepping performs asymptotically worse due
to loss of profile fidelity.
