  The current study aims to evaluate and investigate the development of
projection-based reduced-order models (ROMs) for efficient and accurate RDE
simulations. Specifically, we focus on assessing the projection-based ROM
construction utilizing three different approaches: the linear static basis,
nonlinear quadratic basis, and an adaptive model order reduction (MOR)
formulation. First, an ~\textit{a priori} analysis is performed to evaluate the
effectiveness of the linear static and nonlinear quadratic bases in
representing the detonation-wave dynamics. The~\textit{a priori} analysis
reveals that compared to the linear basis, the nonlinear quadratic basis
provides significantly improved representation of detonation-wave dynamics
within the training regime. However, it exhibits limited capabilities in
representing the dynamics beyond the training regime, either in the future
state or under a different operating parameter (i.e., inlet velocity). Second,
the investigations proceed to the adaptive MOR formulation, which constructs an
\textit{online} adaptive ROM with a small amount of offline training data. It
is demonstrated that the adaptive ROM can provide significantly enhanced
predictive capabilities in modeling the RDE dynamics in the future state, and
subject to parametric variations. More importantly, the adaptive ROM is shown
to be capable of capturing the initial transience in establishing the
detonation wave.
