  This work aims to introduce a heuristic timestep-adaptive algorithm for
Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI)
problems where the flow is dominated by the pressure. In such scenarios, many
time-adaptive algorithms based on the interplay of implicit and explicit time
schemes fail to capture the fast transient dynamics of pressure fields. We
present an algorithm that relies on a temporal error estimator using Backward
Differentiation Formulae (BDF$k$) of order $k=2,3$. Specifically, we
demonstrate that the implicit BDF$3$ solution can be well approximated by
applying a single Newton-type nonlinear solver correction to the implicit
BDF$2$ solution. The difference between these solutions determines our adaptive
temporal error estimator. The effectiveness of our approach is confirmed by
numerical experiments conducted on a backward-facing step flow CFD test case
with Reynolds number $300$ and on a two-dimensional haemodynamics FSI
benchmark.
