A Structure-Preserving JKO Scheme for the Size-Modified Poisson-Nernst-Planck-Cahn-Hilliard Equations

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.