Positivity preserving and mass conservative projection method for the Poisson-Nernst-Planck equation
Fenghua Tong, Yongyong Cai
TL;DR
The paper addresses numerically solving the Poisson-Nernst-Planck equations with positivity preservation and mass conservation at the discrete level. It introduces a two-step scheme consisting of standard time stepping followed by a projection to enforce the physical constraints, using an $L^2$ projection to obtain a Crank-Nicolson-type finite difference discretization that is linear (aside from the projection) and positivity-preserving and mass-conserving. Rigorous $L^2$-norm error estimates establish second-order accuracy in both space and time, and an $H^1$ projection alternative is discussed. Numerical experiments corroborate the theory and illustrate the method's efficiency.
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.