Finite Volumes for a Generalized Poisson-Nernst-Planck System with Cross-Diffusion and Size Exclusion

We present two finite volume approaches for modeling the diffusion of charged particles, specifically ions, in constrained geometries using a degenerate Poisson-Nernst-Planck system with cross-diffusion and volume filling. Both methods utilize a two-point flux approximation and are part of the exponentially fitted scheme framework. The only difference between the two is the selection of a Stolarsky mean for the drift term originating from a self-consistent electric potential. The first version of the scheme, referred to as (SQRA), uses a geometric mean and is an extension of the squareroot approximation scheme. The second scheme, (SG), utilizes an inverse logarithmic mean to create a generalized version of the Scharfetter-Gummel scheme. Both approaches ensure the decay of some discrete free energy. Classical numerical analysis results—existence of discrete solution, convergence of the scheme as the grid size and the time step go to 0—follow. Numerical simulations show that both schemes are effective for moderate Debye lengths, with the (SG) scheme demonstrating greater robustness in the small Debye length limit.