Bound/positivity preserving SAV schemes for the Patlak-Keller-Segel-Navier-Stokes system

The Patlak-Keller-Segel-Navier-Stokes system describes the biological chemotaxis phenomenon in the fluid environment. It is a coupled nonlinear system with unknowns being the cell density, the concentration of chemoattractants, the fluid velocity and the pressure, and it satisfies an energy dissipation law, preserves the bound/positivity and mass of the cell density. We develop in this paper a class of scalar auxiliary variable (SAV) schemes with relaxation which preserve these properties unconditionally at the discrete level, and only require solving decoupled linear systems with constant coefficients at each time step. We present ample numerical results to validate these schemes, simulate the chemotactic non-aggregation and aggregation with a saturation concentration, as well as investigate the blow-up phenomenon.