A Second Order Numerical Scheme of the Cahn-Hilliard-Navier-Stokes System with Flory-Huggins Potential

A second order accurate in time, finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the chemical potential, a modified Crank-Nicolson approximation is applied to the singular logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. Moreover, a nonlinear artificial regularization term is included in the chemical potential approximation, which ensures the positivity-preserving property for the logarithmic arguments, i.e., the numerical value of the phase variable is always between -1 and 1 at a point-wise level. Meanwhile, the convective term in the phase field evolutionary equation is updated in a semi-implicit way, with second order accurate temporal approximation. The fluid momentum equation is also computed by a semi-implicit algorithm. The unique solvability and the positivity-preserving property of the second order scheme is proved, accomplished by an iteration process. A modified total energy stability of the second order scheme is also derived. Some numerical results are presented to demonstrate the accuracy and the robust performance of the proposed second order scheme.