Second-order, Positive, and Unconditional Energy Dissipative Scheme for Modified Poisson-Nernst-Planck Equations

First-order energy dissipative schemes in time are available in literature for the Poisson-Nernst-Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.