Summary: Property analysis is given for nonlinear fully implicit (FI) finite difference discrete scheme with second-order time evolution for conservative diffusion equation. It is proved there exists a unique solution for the nonlinear FI scheme. A Picard-Newton iteration scheme with second-order time accuracy is studied. It is proved the solution of the iteration has second-order convergence both in spatial and temporal variants to the solution of the original problem, and it converges to the solution of the nonlinear discrete scheme with a quadratic speed. The quick solution of the nonlinear problem is realized. The methods here also adapt to analyze first-order time accurate scheme, and can be extended to convection-diffusion problem. Numerical tests verify the high accuracy and efficiency of the second-order temporal evolution Picard-Newton iteration.