An unconditional energy stable data assimilation scheme for Navier-Stokes-Cahn-Hilliard equations with local discretized observed data

In this paper, we introduce the modified Navier-Stokes-Cahn-Hilliard equations with a data assimilation term to utilize the information of the observed data. Based on the idea of feedback control, this term nudges the solutions to the observed data sampled from the reference process. By utilizing the Crank-Nicolson formula and the scalar auxiliary variable approach, we introduce an efficient numerical scheme for the modified Navier-Stokes-Cahn-Hilliard equations. Properties of the mass conservation and unconditional energy stability are proved. We explore the robustness and efficiency of the proposed scheme with various experiments.