A class of positive-preserving,energy stable and high order numerical schemes for the Poission-Nernst-Planck system

TL;DR

This paper develops high-order, energy-stable numerical schemes for the Poisson-Nernst-Planck system that preserve positivity and are verified through theoretical analysis and numerical experiments.

Contribution

It introduces a class of energy variational formulation-based schemes combining semi-implicit time discretization with DG or FE spatial methods, ensuring positivity and stability.

Findings

Schemes are positivity-preserving and energy-stable.

Optimal error estimates and superconvergence are established.

Numerical experiments confirm accuracy and efficiency.

Abstract

In this paper, we introduce and analyze a class of numerical schemes that demonstrate remarkable superiority in terms of efficiency, the preservation of positivity, energy stability, and high-order precision to solve the time-dependent Poisson-Nernst-Planck (PNP) system, which is as a highly versatile and sophisticated model and accommodates a plenitude of applications in the emulation of the translocation of charged particles across a multifarious expanse of physical and biological systems. The numerical schemes presented here are based on the energy variational formulation. It allows the PNP system to be reformulated as a non-constant mobility $H^{-1}$ gradient flow, incorporating singular logarithmic energy potentials. To achieve a fully discrete numerical scheme, we employ a combination of first/second-order semi-implicit time discretization methods, coupled with either the $k$-th order direct discontinuous Galerkin (DDG) method or the finite element (FE) method for spatial discretization. The schemes are verified to possess positivity preservation and energy stability. Optimal error estimates and particular superconvergence results for the fully-discrete numerical solution are established. Numerical experiments are provided to showcase the accuracy, efficiency, and robustness of the proposed schemes.