An energy preserving finite difference scheme for the Poisson-Nernst-Planck system
It provides an efficient and provably stable numerical method for simulating ion transport in electrodiffusion problems.
The paper develops a semi-implicit finite difference scheme for the Poisson-Nernst-Planck system that requires solving only a linear system per time step, achieving second-order spatial and first-order temporal convergence while preserving mass and energy decay.
In this paper, we construct a semi-implicit finite difference method for the time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck system is a nonlinear system, the numerical method presented in this paper only needs to solve a linear system at each time step, which can be done very efficiently. The rigorous proof for the mass conservation and electric potential energy decay are shown. Moreover, mesh refinement analysis shows that the method is second order convergent in space and first order convergent in time. Finally we point out that our method can be easily extended to the case of multi-ions.