Efficient Time-Stepping/Spectral Methods for the Navier-Stokes-Nernst-Planck-Poisson Equations (Communications in Computational Physics, 2018)
This paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poissonequation system. The main goal is to construct and analyzesome stable time stepping schemes for the time discretization and use a spectral methodfor the spatial discretization. The main contribution of the paper includes: 1) an usefulstability inequality for the weak solution is derived; 2) a first order time steppingscheme is constructed, and the non-negativity of the concentration components of thediscrete solution is proved. This is an important property since the exact solutionshares the same property. Moreover, the stability of the scheme is established, togetherwith a stability condition on the time step size; 3) a modified first order scheme is proposedin order to decouple the calculation of the velocity and pressure in the fluid field.This new scheme equally preserves the non-negativity of the discrete concentration solution,and is stable under a similar stability condition; 4) a stabilization technique isintroduced to make the above mentioned schemes stable without restriction conditionon the time step size; 5) finally, we construct a second order finite difference scheme intime and spectral discretization in space. The numerical tests carried out in the papershow that all the proposed schemes possess some desirable properties, such as conditionally/unconditionallystability, first/second order convergence, non-negativity ofthe discrete concentrations, and so on.