The Poisson–Nernst–Planck (PNP) equations describe the dynamics of charged particles in an electric field that is also affected by these particles and have been used to model physical systems involving motions of charged particles, including electrochemistry, semiconductors, and several biological phenomena. When discussing the interplay of an electric field and a flow field, the PNP equation can also be coupled with the Navier–Stokes equation.
In this paper, the authors develop a set of numerical schemes for the Poisson–Nernst–Planck equations. They prove that the schemes are mass conservative, uniquely solvable and keep positivity unconditionally, and that the first-order scheme is energy dissipative unconditionally. These properties hold for various spatial discretizations. Numerical results are presented to validate these properties. Moreover, numerical results indicate that the second-order scheme is also energy dissipative, and both the first- and the second-order scheme preserves the maximum principle for cases where the equation satisfies the maximum principle.
- Numerische Mathematik, 2021, 148(3), 671-697.
- Jie Shen (Purdue University)
- Jie Xu (Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences)