Continuous Data Assimilation Reduced Order Models of Fluid Flow
This work addresses the long-standing problem of ROM inaccuracy over long time intervals and complex flows, offering a practical solution for fluid dynamics simulations.
The authors propose a continuous data assimilation reduced order model (DA-ROM) for incompressible flows, which improves long-time accuracy of ROMs by incorporating measurement data. Numerical tests show the method converges exponentially fast and provides effective long-time accuracy, especially when only inaccurate snapshots are available.
We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend to lose accuracy on more complicated problems and over longer time intervals. Meanwhile, continuous data assimilation (DA) has recently been used to improve accuracy and, in particular, long time accuracy in fluid simulations by incorporating measurement data into the simulation. This paper synthesizes these two ideas, in an attempt to address inaccuracies in ROM by applying DA, especially over long time intervals and when only inaccurate snapshots are available. We prove that with a properly chosen nudging parameter, the proposed DA-ROM algorithm converges exponentially fast in time to the true solution, up to discretization and ROM truncation errors. Finally, we propose a strategy for nudging adaptively in time, by adjusting dissipation arising from the nudging term to better match true solution energy. Numerical tests confirm all results, and show that the DA-ROM strategy with adaptive nudging can be highly effective at providing long time accuracy in ROMs.