Efficient, Positive, and Energy Stable Schemes for Multi-D Poisson–Nernst–Planck Systems

In this paper, we design, analyze, and numerically validate positive energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson–Nernst–Planck equations, which has found much use in modeling biological membrane channels semiconductor devices. The semi-implicit time discretization based on a reformulation gives well-posed elliptic system, is shown to preserve solution positivity arbitrary steps. first order (in time) fully-discrete scheme mass conservation unconditionally, energy dissipation with only mild O(1) step restriction. also steady-states. For fully second both space) large steps, restored by local scaling limiter, maintain spatial accuracy. These are easy implement. Several three-dimensional numerical examples verify our theoretical findings demonstrate accuracy, efficiency, robustness proposed schemes, as well fast approach