Abstract
In this work, we propose a Keller-Segel-Navier-Stokes (KSNS) model for describing chemotactic phenomena, formulated within the framework of the Energetic Variational Approach (EnVarA). A second-order accurate numerical scheme is developed that rigorously preserves three fundamental properties in discrete sense: the positivity of cell density, mass conservation of cell density, and total energy dissipation. The Keller-Segel subsystem is reformulated as a coupling between an H-1 gradient flow with non-constant mobility and an L-2 gradient flow, enabling the effective treatment of the nonlinear and singular logarithmic energy potential via a modified Crank-Nicolson scheme. Artificial regularization terms are introduced to enforce positivity preservation. For the fluid dynamics component, we adopt a second-order semi-implicit time discretization. The marker-and-cell (MAC) finite difference approximation is used as the spatial discretization, which ensures a discretely divergence-free velocity field. The proposed numerical method guarantees unique solvability, mass conservation, and total energy stability. Furthermore, through detailed asymptotic expansions and rigorous error analysis, we establish optimal convergence rates. A series of numerical experiments are presented to validate the effectiveness and robustness of both the physical model and the numerical scheme.