A conservative discretization of the Poisson-Nernst-Planck equations on adaptive Cartesian grids

In this paper we present a novel hybrid finite-difference/finite-volume method for the numerical solution of the nonlinear Poisson-Nernst-Planck (PNP) equations on irregular domains. The method is described in two spatial dimensions but can be extended to three dimensional problems as well. The boundary of the irregular domain is represented implicitly via the zero level set of a signed distance function and quadtree data structures are used to systematically generate adaptive grids needed to accurately capture the electric double layer near the boundary. To handle the nonlinearity in the PNP equations efficiently, a semi-implicit time integration method is utilized. An important feature of our method is that total number of ions in the system is conserved by carefully imposing the boundary conditions, by utilizing a conservative discretization of the diffusive and, more importantly, the nonlinear migrative flux term. Several numerical experiments are conducted which illustrate that the presented method is first-order accurate in time and second-order accurate in space. Moreover, these tests explicitly indicate that the algorithm is also conservative. Finally we illustrate the applicability of our method in the study of the charging dynamics of porous supercapacitors.

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