The Voronoi box based two point flux finite volume method provides a path to discretization approaches for systems of partial differential equations which conform to natural constraints of solutions and basic principles of thermodynamics.  As a case in point, the talk introduces an adaptation of the well-known Scharfetter-Gummel upwind scheme from semiconductor physics to generalized Nernst-Planck-Poisson systems taking into accout finite ion size and solvation effects ^[1].

The method has been implemented in the Julia programming language  using the package VoronoiFVM.jl ^[2].  It provides solution methods for coupled nonlinear reaction-convection-diffusion problems in one-, two- and three-dimensional spatial domains.  A key ingredient of this package is the utilization if automatic differentiation to tackle complex nonlinearities in realistic physical models, see e.g. ^[3,4,5].

We will discuss a number of work-in-progress examples demonstrating the utility of this approach in the context of electrolyte simulations including
- Model problem based simulation of double layer effects on electrochemical reactions
- Calculation of electroosmotic flows by coupling with pressure robust finite element methods (Julia based re-implementation of the approach in ^[6])
- Automatic generation of reaction terms from chemical equations using Catalyst.jl <sup>[7]


[1] B. Gaudeul and J. Fuhrmann, "Entropy and convergence analysis for two finite volume schemes for a Nernst-Planck-Poisson system with ion volume constraints", Numerische Mathematik, vol. 151, no. 1, pp. 99–149, 2022
[2] J. Fuhrmann and contributors, https://urldefense.com/v3/__https://github.com/WIAS-PDELib/VoronoiFVM.jl__;!!Mih3wA!DMZTW98Az4xv1B69eOyiSWXUGZQn0qdiyc21NPoVpmuEkF-9hHD07uK0NO4d1mPc84rpGuX0fuTTmJ0Bp9cROjRyOD0C-vQ46g$
[3] Ch. Keller, J. Fuhrmann, and M. Landstorfer, "A model framework for ion channels with selectivity filters based on continuum non-equilibrium thermodynamics", Entropy 2025, 27(9), 981
[4] V. Miloš, P. Vágner, D. Budáč, M. Carda, M. Paidar, J. Fuhrmann, and K. Bouzek, "Generalized Poisson-Nernst-Planck-based physical model of an O2 | LSM | YSZ electrode", Journal of the Electrochemical Society,  no. 169, p. 044505, 2022
[5] D. Brust, K. Hopf, A. Cheilytko, M. Wullenkord, and Ch. Sattler, "Transport of heat and mass for reactive gas mixtures in porous media: Modeling and application", Chemical Engineering Journal 516(15) 2025, 162027
[6] J. Fuhrmann, C. Guhlke, A. Linke, C. Merdon, and R. Müller, “Induced charge electroosmotic flow with finite ion size and solvation effects,” Electrochimica Acta, vol. 317, pp. 778–785, 2019
[7] Loman, T. E., Ma, Y., Ilin, V., Gowda, S., Korsbo, N., Yewale, N., Rackauckas, Ch & Isaacson, S. A. (2023). Catalyst: Fast and flexible modeling of reaction networks. PLOS Computational Biology, 19(10), e1011530.</sup>