期刊
JOURNAL OF SCIENTIFIC COMPUTING
卷 87, 期 3, 页码 -出版社
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-021-01503-1
关键词
Poisson– Nernst– Planck equations; Semi-implicit discretization; Mass conservation; Positivity; Energy decay; Steady-state
资金
- National Science Foundation [DMS1812666]
This paper presents positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck equations, demonstrating their accuracy, efficiency, and robustness through numerical examples.
In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully-discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O(1) time step restriction. The scheme is also shown to preserve the steady-states. For the fully second order (in both time and space) scheme with large time steps, solution positivity is restored by a local scaling limiter, which is shown to maintain the spatial accuracy. These schemes are easy to implement. Several three-dimensional numerical examples verify our theoretical findings and demonstrate the accuracy, efficiency, and robustness of the proposed schemes, as well as the fast approach to steady-states.
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