Convergence Analysis on a Structure-Preserving Numerical Scheme for the Poisson-Nernst-Planck-Cahn-Hilliard System (CSIAM Transactions on Applied Mathematics, 2023)
In this paper, we provide an optimal rate convergence analysis and errorestimate for a structure-preserving numerical scheme for the Poisson-Nernst-Planck-Cahn-Hilliard (PNPCH) system. The numerical scheme is based on the Energetic Variational Approach of the physical model, which is reformulated as a non-constant mobility gradient flow of a free-energy functional that consists of singular logarithmicenergy potentials arising from the PNP theory and the Cahn-Hilliard surface diffusion process. The mobility function is explicitly updated, while the logarithmic andthe surface diffusion terms are computed implicitly. The primary challenge in the development of theoretical analysis on optimal error estimate has been associated withthe nonlinear parabolic coefficients. To overcome this subtle difficulty, an asymptoticexpansion of the numerical solution is performed, so that a higher order consistencyorder can be obtained. The rough error estimate leads to a bound in maximum normfor concentrations, which plays an essential role in the nonlinear analysis. Finally, therefined error estimate is carried out, and the desired convergence estimate is accomplished. Numerical results are presented to demonstrate the convergence order andperformance of the numerical scheme in preserving physical properties and capturingionic steric effects in concentrated electrolytes.
Keywords: Poisson-Nernst-Planck-Cahn-Hilliard system; positivity preserving; optimal rate convergence analysis; higher order asymptotic expansion; rough error estimate; refined error estimate.