Analysis and computation of continuous data assimilation algorithms for Lorenz 63 system based on nonlinear nudging techniques

Abstract

In this work, continuous data assimilation algorithms based on three nonlinear nudging techniques (simple power nonlinearity, hybrid linear–nonlinear method and concave–convex nonlinearity) are studied for Lorenz 63 system to speed up the convergence rate of the approximate solution to the reference solution. The well-posedness of these three nonlinear continuous data assimilation algorithms are proven. For two cases that include synchronizing the first or second variable only are considered, the approximate solution is proven to converge to the reference solution as times go to infinity provided that the relaxed parameter related to the corresponding variable is sufficiently large. The byproduct of the proof is to observe that for these three nonlinear nudging techniques, synchronizing the second variable could be more efficient than synchronizing the first variable because the convergence rate for the first variable is shown to be slower than the second variable. Although the convergence rate may not be optimal, this still gives some possible explanation why synchronizing the second variable is more efficient than synchronizing the first which are numerically found in our numerical simulations. Moreover, numerical simulations are presented to illustrate the advantages of continuous data assimilation algorithms based on nonlinear nudging techniques. Some open problems arising from numerical experiments are reported.