We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of the analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a result about convergence. The main result shows convergence of consistent finite difference schemes even without stability, and therefore shows independence between stability and convergence for finite difference schemes. Our theoretical result can be realized numerically on multiple-precision arithmetic environments.