The Nernst-Planck-Poisson equations (NPP) are used to describe transport of ions in electrochemical and biological membranes. The nature of electrochemical processes involve critical phenomena near the boundary of the membrane (at distances of the Debye length) while the thickness of the membrane can vary up to milimeters. Different range of lengths from nano- to microscales implies difficulties in numerical simulations. Effective solutions of the time-dependent (transient-state) and time-independent (steady-state) NPP problem are presented for one dimension geometry.

The time-dependent form of the Nernst-Planck-Poisson equations can be used both for transient and steady-state calculations. Steady-state analysis is obtained by starting from an initial profiles, and letting the numerical system evolve until a stationary solution is reached. However it is not always obvious if the state, which has been achieved, is really the steady-state. We have devised a method for verification whether time-dependent NPP system reached its steady-state. This approach requires the solving a steady-state NPP problem with Dirichlet type boundary conditions.

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