On the Nudging Approach to Continuous Data Assimilation in the Limit of Infinite Error Feedback Gain

Abstract. This article studies the intimate relationship between two filtering algorithms for continuous data assimilation, the direct-replacement algorithm and the nudging algorithm, in the paradigmatic context of the two-dimensional (2D) Navier–Stokes equations (NSE) for incompressible fluids. In this setting, the nudging algorithm can formally be viewed as an affine perturbation of the 2D NSE. Thus, in the degenerate limit of zero nudging parameter, the nudging algorithm correspondingly converges to the solution of the 2D NSE. However, when the nudging parameter of the nudging algorithm is made increasingly large, the perturbation apparently becomes more singular. It is nevertheless shown here that in the limit of infinite nudging parameter, the nudging filter converges to the direct-replacement algorithm. In establishing this result, the article fills a notable gap in the literature surrounding these algorithms. Numerical experiments are then presented that confirm the theoretical results and probe the issue of selecting a nudging strategy in the presence of observational noise. In this direction, an adaptive nudging strategy is proposed that leverages the insight gained from the relationship between the direct-replacement algorithm and the nudging algorithm that produces measurable improvement over the constant nudging strategy.–