| Abstract: |
| In this paper, we present a class of efficient, positive-preserving, energy stable and high order numerical schemes are presented and studied for solving the time-dependent Poisson-Nernst-Planck (PNP) system. The numerical scheme is based on the energy variational formulation and the PNP system is reformulated as a non-constant mobility $H^1$ gradient flow, with singular logarithmic energy potentials involved. The fully discrete numerical scheme is constructed by using the first/second order semi-implicit time discretization coupled with the $k$-th order direct discontinuous Galerkin (DDG) method or the finite element (FE) method for space discretization. The scheme is shown to be positivity preserving and energy stable. Furthermore, optimal error estimates and some superconvergence results are established for the fully-discrete numerical solution. Numerical experiments are provided to demonstrate the accuracy, efficiency, and robustness of the proposed scheme. |
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