Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study (Communications in Computational Physics, 2021)
Data assimilation is a technique for increasing the accuracy of simulationsof solutions to partial differential equations by incorporating observable data into thesolution as time evolves. Recently, a promising new algorithm for data assimilationbased on feedback-control at the PDE level has been proposed in the pioneering workof Azouani, Olson, and Titi (2014). The standard version of this algorithm is based onmeasurement from data points that are fixed in space. In this work, we consider thescenario in which the data collection points move in space over time. We demonstratecomputationally that, at least in the setting of the 1D Allen-Cahn reaction diffusionequation, the algorithm converges with significantly fewer measurement points, up toan order or magnitude in some cases. We also provide an application of the algorithmto the estimation of a physical length scale in the case of a uniform static grid.
Keywords: Continuous data assimilation; Allen-Cahn; reaction-diffusion; moving mesh; synchronization.