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We study different approaches to implementing sparse-in-time observationsinto the the Azouani-Olson-Titi data assimilation algorithm. We propose a newmethod which introduces a "data assimilation window" separate from theobservational time interval. We show that by making this window as small aspossible, we can drastically increase the strength of the nudging parameterwithout losing stability. Previous methods used old data to nudge the solutionuntil a new observation was made. In contrast, our method stops nudging thesystem almost immediately after an observation is made, allowing the systemrelax to the correct physics. We show that this leads to an order-of-magnitudeimprovement in the time to convergence in our 3D Navier-Stokes simulations.Moreover, our simulations indicate that our approach converges at nearly thesame rate as the idealized method of direct replacement of low Fourier modesproposed by Hayden, Olson, and Titi (HOT). However, our approach can be readilyadapted to non-idealized settings, such as finite element methods, finitedifference methods, etc., since there is no need to access Fourier modes as ourmethod works for general interpolants. It is in this sense that we think of ourapproach as ``second best;'' that is, the ``best'' method would be the directreplacement of Fourier modes as in HOT, but this idealized approach istypically not feasible in physically realistic settings. While our method has aconvergence rate that is slightly sub-optimal compared to the idealized method,it is directly compatible with real-world applications. Moreover, we proveanalytically that these new algorithms are globally well-posed, and converge tothe true solution exponentially fast in time. In addition, we provide the first3D computational validation of HOT algorithm.

The second-best way to do sparse-in-time continuous data assimilation: Improving convergence rates for the 2D and 3D Navier-Stokes equations