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First-order energy dissipative schemes in time are available in literaturefor the Poisson-Nernst-Planck (PNP) equations, but second-order ones are stillin lack. This work proposes novel second-order discretization in time andfinite volume discretization in space for modified PNP equations thatincorporate effects arising from ionic steric interactions and dielectricinhomogeneity. A multislope method on unstructured meshes is proposed toreconstruct positive, accurate approximations of mobilities on faces of controlvolumes. Numerical analysis proves that the proposed numerical schemes are ableto unconditionally ensure the existence of positive numerical solutions,original energy dissipation, mass conservation, and preservation of steadystates at discrete level. Extensive numerical simulations are conducted todemonstrate numerical accuracy and performance in preserving properties ofphysical significance. Applications in ion permeation through a 3D nanoporeshow that the modified PNP model, equipped with the proposed schemes, haspromising applications in the investigation of ion selectivity andrectification. The proposed second-order discretization can be extended todesign temporal second-order schemes with original energy dissipation for atype of gradient flow problems with entropy.

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