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.

在本文中，我们设计，分析并数值验证了正和耗能方案，以解决与时间有关的泊松-能斯特-普朗克方程的多维系统，该系统在生物膜通道和半导体的建模中发现了许多用途设备。基于系统重构的半隐式时间离散化给出了一个位置良好的椭圆系统，该系统被证明可以在任意时间步长下保持解的正性。所示的一阶（及时）全离散方案可无条件地保持溶液的正性和质量守恒，并且能量消耗仅需适度的O（1）时间步长的限制。该方案还显示出保持稳态。对于具有较大时间步长的完全二阶（在时间和空间上）方案，通过局部缩放限制器可以恢复解的正性，这表明可以保持空间精度。这些方案易于实现。几个三维数值示例验证了我们的理论发现，并证明了所提出方案的准确性，效率和鲁棒性，以及快速实现稳态的方法。