Energy dissipative and positivity preserving schemes for large-convection ion transport with steric and solvation effects

Ion transport plays a crucial role in biophysical and electrochemical applications. While being modeled by the classical Poisson-Nernst-Planck (PNP) equations, many ionic features are ignored in the course-graining modeling of ionic interactions. In this work, we study ion transport by modified PNP equations including effects arising from steric interactions and the Born solvation, which both could cause strong convection. The modified NP equations are first rewritten in a gradient-flow reformulation that involves mobility and chemical potential terms. The mobility is then approximated by upwind fluxes with various flux limiters on the half-grid points, and the chemical potential terms are then implicitly approximated by central differencing. The resulting first-order in time scheme is proved to be energy dissipative unconditionally. Furthermore, the unconditional existence of positive numerical concentration is proved by making use of the singularity of the entropy term, which prevents the concentration from approaching zero. The flux with the Minmod or Osher limiter is reduced to a second-order upwind flux in the strong convective limit. Numerical tests are performed to confirm the expected order of accuracy and the properties of the developed scheme at discrete level. In addition, our numerical experiments demonstrate that the developed schemes with limiters can achieve higher accuracy than the Scharfetter-Gummel scheme and classical upwind scheme in tests with strong convection. Finally, the developed schemes are applied to numerically assess the effects of Coulombic interaction, steric interaction, and Born solvation on the rectifying transport of ions through nanopores.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this study, ion transport was investigated by modifying the PNP equations to account for steric interactions and Born solvation effects that can cause strong convection. The modified equations were solved using a first-order in time scheme that is proven to be energy dissipative unconditionally. Numerical tests confirmed the accuracy and properties of the developed scheme, showing higher accuracy than conventional schemes in tests with strong convection.