On an electro-neutral model of dynamic Poisson-Nernst-Planck system

A systematic asymptotic study is conducted for the dynamic Poisson-Nernst-Planck (PNP) system, that describes the transport of ions. One intriguing feature of this system is the presence of a thin boundary/Debye layer, and this presents significant challenges on numerical computation and on deriving macroscopic continuum models for cellular structures from microscopic description. This work mainly focuses on the one-dimensional system, including the general multi-ion case. We manage to derive an electro-neutral (EN) system for the bulk region, with a variety of effective boundary conditions reduced from original ones. Then, this EN system can be solved directly and efficiently without calculating the solution in boundary layer. The derivation is based on matched asymptotics, and the key idea is to bring back some higher order contributions into effective boundary conditions. For flux boundary conditions, these effective conditions are physically incorrect to omit such contributions, which account for accumulation of ions in boundary layer; while for Dirichlet boundary conditions, they can be considered as generalization of continuity of electrochemical potential. The validity of the EN system is verified by a number of numerical examples. Particularly, we have derived an EN model for neuronal axon by combining with the Hodgkin-Huxley model, and demonstrated the phenomenon of action potential. The computational time is significantly reduced, since the it needs less mesh points and allows relatively large time step.