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We have developed efficient numerical algorithms for solving 3D steady-statePoisson-Nernst-Planck (PNP) equations with excess chemical potentials describedby the classical density functional theory (cDFT). The coupled PNP equationsare discretized by a finite difference scheme and solved iteratively using theGummel method with relaxation. The Nernst-Planck equations are transformed intoLaplace equations through the Slotboom transformation. Then, the algebraicmultigrid method is applied to efficiently solve the Poisson equation and thetransformed Nernst-Planck equations. A novel strategy for calculating excesschemical potentials through fast Fourier transforms is proposed, which reducescomputational complexity from $O(N^2)$ to $O(N\log N)$, where $N$ is the numberof grid points. Integrals involving the Dirac delta function are evaluateddirectly by coordinate transformation, which yields more accurate resultscompared to applying numerical quadrature to an approximated delta function.Numerical results for ion and electron transport in solid electrolyte forlithium-ion (Li-ion) batteries are shown to be in good agreement with theexperimental data and the results from previous studies.

communications in computational physics

Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with  Classical Density Functional Theory for Modeling Ion and Electron Transport  in a Confined Environment