A linear, second-order accurate, positivity-preserving and unconditionally energy stable scheme for the Navier-Stokes-Poisson-Nernst-Planck system

Electrokinetic phenomena involving the transport of ions in fluids generally appears in biological system, hydrodynamics and geodynamics. The underlining mathematical models involve the nonlinear coupling between the fluid motion and ion transportation. Performing efficient, accurate and structure preserving computation for these electrohydrodynamic models is challenging due to their multiphysics and complex nature. The Navier-Stokes-Poisson-Nernst- Planck (NSPNP) system is a widely used electrokinetic model in the existing literature. In this paper, a linear, second -order accurate, positivity -preserving and unconditionally energy stable scheme is developed for the NSPNP system. First, a square root transformation is employed to ensure the positivity of ion concentrations. Next, a specific ordinary differential equation is introduced to deal with the nonlinear coupling terms satisfying the "zero-energy-contribution"feature, which is combined with the scalar auxiliary variable method for nonlinear potentials to achieve a decoupled energy stable scheme. Therefore, only several linearly independent equations need to be solved at each time step. The staggered grid finite difference method is used for the spatial discretizations, and the unconditional energy stability of the fully discrete scheme is established. Finally, the unique solvability of the scheme is proved. Numerical experiments are carried out to demonstrate the accuracy and performance of the proposed scheme, and the effect of electric Coulomb force on the flow and motion of ions are also illustrated.