Efficient numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equations

TL;DR

This paper introduces efficient, unconditionally stable first- and second-order semi-implicit schemes for the coupled Navier-Stokes-Nernst-Planck-Poisson equations, ensuring positivity, mass conservation, and computational efficiency.

Contribution

It presents the first second-order method that guarantees stability, positivity, and mass conservation for these complex coupled equations.

Findings

Schemes are unconditionally stable with energy decay.

Concentration components preserve positivity and mass.

Numerical examples confirm accuracy and efficiency.

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 reformulation and sophisticated treatment of the terms coupling different equations. 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 advantages of the proposed method are: (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. Some numerical examples are presented to demonstrate the accuracy and performance of the proposed method. As far as the best we know, this is the first second-order method which satisfies all the above properties for the Navier-Stokes-Nernst-Planck-Poisson equations.