A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson-Nernst-Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst-Planck equations, which was developed in [J. Sci. Comput. 81(2019),436-458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove firstorder convergence in time and second-order convergence in space for the numerical scheme in the l(infinity) (0, Y; l(2)) boolean AND l(2)(0, T; H-h(1)) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an l(infinity) bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.