  We have developed efficient numerical algorithms for solving 3D steady-state
Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described
by the classical density functional theory (cDFT). The coupled PNP equations
are discretized by a finite difference scheme and solved iteratively using the
Gummel method with relaxation. The Nernst-Planck equations are transformed into
Laplace equations through the Slotboom transformation. Then, the algebraic
multigrid method is applied to efficiently solve the Poisson equation and the
transformed Nernst-Planck equations. A novel strategy for calculating excess
chemical potentials through fast Fourier transforms is proposed, which reduces
computational complexity from $O(N^2)$ to $O(N\log N)$, where $N$ is the number
of grid points. Integrals involving the Dirac delta function are evaluated
directly by coordinate transformation, which yields more accurate results
compared to applying numerical quadrature to an approximated delta function.
Numerical results for ion and electron transport in solid electrolyte for
lithium-ion (Li-ion) batteries are shown to be in good agreement with the
experimental data and the results from previous studies.
