Almost Newton method for large flux steady-state of 1D Poisson–Nernst–Planck equations

Problems of charge-carrier transport from many different fields converge in mathematics, where they are modeled by a system of Poissonʹs and Nernst–Planckʹs equations (PNP) for the electro-static potential and particle dynamics. In this paper, we study a computational steady-state problem of charge-carrier transport. In the case of Dirichlet boundary conditions for the electro-static potential at all locations with given particle densities, the Gummel method (IEEE Trans. Electron Devices 11 (1964) 455) is known to converge to the steady-state solution rapidly, and to high accuracy, so long as the steady-state flux densities remain small. We wish to predict the far from equilibrium, large flux, steady-state of a charged particle system with two compartments, separated by a semi-permeable membrane, in which the electro-static potential is unknown at all but one location with given particle densities. In this case all but one Dirichlet boundary conditions on the electro-static potential are replaced by Neumann boundary conditions. We derive a modified Gummel method (MG) capable of solving such a Dirichlet–Neumann boundary problem. We investigate its difficulties with large steady-state flux densities by comparing it to the full Newton method (FN). Since problems with FN at large steady-state flux densities have been reported (IEEE Trans. Comput. Aid D 7 (2) (1988) 251; J. Appl. Phys. 68 (3) (1990) 1324), we propose, derive, and compare to MG and FN, an almost Newton method (AN) based upon a partial linearization of the problem. AN achieves the same accuracy as MG and FN in only one fifth of the number of iteration steps and retains these qualities even at very large steady-state flux densities.