TY - JOUR
T1 - A Structure-Preserving JKO Scheme for the Size-Modified Poisson-Nernst-Planck-Cahn-Hilliard Equations
AU - Ding , Jie
AU - Ji , Xiang
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 204
EP - 229
PY - 2023
DA - 2023/01
SN - 16
DO - http://doi.org/ 10.4208/nmtma.OA-2022-0078
UR - https://global-sci.org/intro/article_detail/nmtma/21349.html
KW - Structure-preserving, size-modified Poisson-Nernst-Planck-Cahn-Hilliard equations, JKO framework, positivity.
AB - <p style="text-align: justify;">In this paper, we propose a structure-preserving numerical scheme for
the size-modified Poisson-Nernst-Planck-Cahn-Hilliard (SPNPCH) equations derived
from the free energy including electrostatic energies, entropies, steric energies, and
Cahn-Hilliard mixtures. Based on the Jordan-Kinderlehrer-Otto (JKO) framework
and the Benamou-Brenier formula of quadratic Wasserstein distance, the SPNPCH
equations are transformed into a constrained optimization problem. By exploiting
the convexity of the objective function, we can prove the existence and uniqueness of the numerical solution to the optimization problem. Mass conservation and
unconditional energy-dissipation are preserved automatically by this scheme. Furthermore, by making use of the singularity of the entropy term which keeps the
concentration from approaching zero, we can ensure the positivity of concentration.
To solve the optimization problem, we apply the quasi-Newton method, which can
ensure the positivity of concentration in the iterative process. Numerical tests are
performed to confirm the anticipated accuracy and the desired physical properties
of the developed scheme. Finally, the proposed scheme can also be applied to study
the influence of ionic sizes and gradient energy coefficients on ion distribution.</p>