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arXiv · 2403.04291

Positivity preserving and mass conservative projection method for the Poisson-Nernst-Planck equation

Fenghua Tong·Yongyong Cai
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Abstract

We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the $L^2$ projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient $L^2$ projection part), positivity preserving and mass conserving. Rigorous error estimates in $L^2$ norm are established, which are both second order accurate in space and time. The other choice of projection, e.g. $H^1$ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.

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Fenghua Tong, Yongyong Cai. 2024-03-07. Positivity preserving and mass conservative projection method for the Poisson-Nernst-Planck equation. https://arxiv.org/abs/2403.04291

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