Discrete‐in‐Time Data Assimilation for Reaction–Diffusion Models Using the Delay Sparse Data

ABSTRACT In this paper, we propose a new algorithm for predicting future states of the semilinear reaction–diffusion models with unknown or incomplete initial data in science and engineering by solely using the sparse data from the past. This algorithm is based on the discrete approximation of the feedback control (nudging) approach with the measurements given on a coarse mesh by implicit–explicit stage‐based interpolation Runge–Kutta (RK) method for time discretization and finite element method for the spatial discretization. The stability of the time semi‐discrete and fully discrete data assimilation approximations to the models is first shown under suitable conditions on the nudging parameter by exploring the algebraical stability of the RK methods. The error estimates derived for the state prediction algorithm demonstrate that the time semi‐discrete and fully discrete approximations converge to the true state exponentially over time. Several numerical examples are provided to show the efficiency of this proposed algorithm for predicting future states.