  The vast majority of reduced-order models (ROMs) first obtain a low
dimensional representation of the problem from high-dimensional model (HDM)
training data which is afterwards used to obtain a system of reduced
complexity. Unfortunately, convection-dominated problems generally have a
slowly decaying Kolmogorov n-width, which makes obtaining an accurate ROM built
solely from training data very challenging. The accuracy of a ROM can be
improved through enrichment with HDM solutions; however, due to the large
computational expense of HDM evaluations for complex problems, they can only be
used parsimoniously to obtain relevant computational savings. In this work, we
exploit the local spatial and temporal coherence often exhibited by these
problems to derive an accurate, cost-efficient approach that repeatedly
combines HDM and ROM evaluations without a separate training phase. Our
approach obtains solutions at a given time step by either fully solving the HDM
or by combining partial HDM and ROM solves. A dynamic sampling procedure
identifies regions that require the HDM solution for global accuracy and the
reminder of the flow is reconstructed using the ROM. Moreover, solutions
combining both HDM and ROM solves use spatial filtering to eliminate potential
spurious oscillations that may develop. We test the proposed method on inviscid
compressible flow problems and demonstrate speedups up to an order of
magnitude.
