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

We study different approaches to implementing sparse-in-time observations into the the Azouani-Olson-Titi data assimilation algorithm. We propose a new method which introduces a "data assimilation window" separate from the observational time interval. We show that by making this window as small as possible, we can drastically increase the strength of the nudging parameter without losing stability. Previous methods used old data to nudge the solution until a new observation was made. In contrast, our method stops nudging the system almost immediately after an observation is made, allowing the system relax to the correct physics. We show that this leads to an order-of-magnitude improvement in the time to convergence in our 3D Navier-Stokes simulations. Moreover, our simulations indicate that our approach converges at nearly the same rate as the idealized method of direct replacement of low Fourier modes proposed by Hayden, Olson, and Titi (HOT). However, our approach can be readily adapted to non-idealized settings, such as finite element methods, finite difference methods, etc., since there is no need to access Fourier modes as our method works for general interpolants. It is in this sense that we think of our approach as ``second best;'' that is, the ``best'' method would be the direct replacement of Fourier modes as in HOT, but this idealized approach is typically not feasible in physically realistic settings. While our method has a convergence rate that is slightly sub-optimal compared to the idealized method, it is directly compatible with real-world applications. Moreover, we prove analytically that these new algorithms are globally well-posed, and converge to the true solution exponentially fast in time. In addition, we provide the first 3D computational validation of HOT algorithm.