A Positivity Preserving and Free Energy Dissipative Difference Scheme for the Poisson-Nernst-Planck System

In this paper, based on the reformulation of the Nernst-Planck equations, we construct an unconditionally stable semi-implicit linearized difference scheme for the time dependent Poisson-Nernst-Planck system. The method preserves several important physical laws at full discrete level without any constraints on the time step size, including: mass conservation, ion concentration positivity, free energy dissipation. The proposed method is a linearized method, which can be implemented efficiently. Furthermore, the method can be extended to the case of multi-ions without any difficulty. The paper ends with supporting numerical results which confirm well with theoretical findings.