Positivity Preserving and Mass Conservative Projection Method for the Poisson–Nernst–Planck Equation

.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 the \(L^2\) 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.