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

2024/03/07 by Fenghua Tong, Tong, Fenghua, Yongyong Cai +1 · 2 citations
Mathematics · #Gas Dynamics and Kinetic Theory #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2403.04291

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 L² projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient L² projection part), positivity preserving and mass conserving. Rigorous error estimates in L² norm are established, which are both second order accurate in space and time. The other choice of projection, e.g. H¹ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.

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• Efficient structure-preserving scheme for chemotaxis PDE system with singular sensitivity in crime and epidemic modeling
• Positive-preserving, mass conservative linear schemes for the Possion-Nernst-Planck equations

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