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.

翻译：本文提出并分析了一种构建泊松-能斯特-普朗克方程结构保持近似的新方法，重点聚焦于正性保持和质量守恒特性。该策略由标准时间推进步和投影（修正）步组成，以满足所需的物理约束（正性和质量守恒）。基于$L^2$投影，我们构建了一种二阶Crank-Nicolson型有限差分格式，该格式为线性（高效的$L^2$投影部分除外），且具有正性保持和质量守恒性。建立了严格的$L^2$范数误差估计，其在空间和时间上均达到二阶精度。还讨论了其他投影选择（如$H^1$投影）。通过数值算例验证了理论结果并展示了所提方法的有效性。