A second-order discretization of the nonlinear Poisson-Boltzmann equation over irregular geometries using non-graded adaptive Cartesian grids

In this paper we present a nite di erence scheme for the discretization of the nonlinear PoissonBoltzmann(PB) equation over irregular domains that is second-order accurate. The interface is represented by a zero level set of a signed distance function using Octree data structure, allowing a natural and systematic approach to generate non-graded adaptive grids. Such a method guaranties computational e ciency by ensuring that the nest level of grid is located near the interface. The nonlinear PB equation is discretized using nite di erence method and several numerical experiments are carried which indicate the second-order accuracy of method. Finally the method is used to study the supercapacitor behaviour of porous electrodes.