Finite time synchronization of the continuous/discrete data assimilation algorithms for Lorenz 63 system based on the back and forth nudging techniques

In this work, we numerically and theoretically study the finite time synchronization of the continuous/discrete data assimilation algorithms for Lorenz 63 system based on the back and forth nudging (BFN) techniques. The proof of the continuous data assimilation algorithm is given under the assumption that the coupling parameter is larger than a constant depending on the reference solution. To adjust the realistic situation that the observational measurement is collected discretely in time, the discrete data assimilation algorithm is also studied and the convergence of the algorithm is proven under the assumption that the assimilated time step is smaller than a constant depending on the reference solution. Finally, several numerical experiments with linear and nonlinear back and forth nudging techniques are presented to confirm the theoretical results studied in this work and also ensure the robustness of the algorithm.