Efficient time-stepping schemes for the Navier-Stokes-Nernst-Planck-Poisson equations

Abstract

We propose in this paper efficient first/second-order time-stepping schemes for the evolutional Navier-Stokes-Nernst-Planck-Poisson equations. The proposed schemes are constructed using an auxiliary variable approach. By introducing a dynamic equation for the auxiliary variable and reformulating the original equations into an equivalent system, we construct first- and second-order semi-implicit linearized schemes for the underlying problem. The main contribution of the paper includes: (1) the schemes are unconditionally stable in the sense that a discrete energy keeps decay during the time stepping; (2) the concentration components of the discrete solution preserve positivity and mass conservation; (3) the delicate implementation shows that the proposed schemes can be very efficiently realized, with computational complexity close to a semi-implicit scheme. As far as the best we know, it is the first time for constructing a second-order method which satisfies all the above properties for the Navier-Stokes-Nernst-Planck-Poisson equations. Some numerical examples are presented to demonstrate the accuracy and performance of the proposed method.