Almost Newton method for large flux steady-state of 1D Poisson--Nernst--Planck equations (Q2488283)

Problems of charge-carrier transport from many different fields converge in mathematics, where they are modeled by a system of Poisson's and Nernst-Planck's equations (PNP) for the electrostatic potential and the particle dynamics. In the present paper, a computational steady-state problem of charge-carrier transport is considered, where particle densities are fixed at values other than the equilibrium values and fluxes are non-zero. In the case of Dirichlet boundary conditions for the electrostatic potential at all locations with given particle densities, the Gummel method [IEEE Trans. Electron Devices 11, 455 ff (1964)] is known to converge to the steady-state solution rapidly, and to high accuracy, so long as the steady-state flux densities remain small. Here the author studies the steady state of charged particle systems far from equilibrium at large fluxes. The systems consist of two compartments separated by a semi-permeable membrane, in which the electrostatic potential is unknown at all but one location of known particle densities. In this case all but one Dirichlet boundary conditions on the electrostatic potential are replaced by Neumann boundary conditions. The author derives a modified Gummel method (MG) capable of solving such a Dirichlet-Neumann boundary problem. Besides, the full linearization of the PNP equation is computed, the discretization of which defines the full Newton iteration (FN). But, as in the literature problems with FN at large steady-state flux densities are reported [IEEE Trans. Comput. Aid D 7, No. 2, 251 ff (1988)], further an almost Newton method (AN) is proposed, which is based upon a partial linearization of the problem. Finally, numerical solutions are obtained by the three methods MG, FN and AN, and they are compared. It is found that AN and FN require a high effort to obtain the solution at each operation step compared to MG. However, AN has been shown to avoid FN's problems with catastrophic cancellations as well as MG's sensitivity to flux densities that stems from neglecting them. AN converges with about one fifth of the number of iteration steps to the same accuracy as MG and FN and is insensitive to changes in flux densities. AN also converges faster than FN and MG in absolute time which implies that it needs fewer flops. Therefore, AN is an excellent candidate for the simulation of consecutive steady-state dynamics between two bulk compartments of the system.