  This paper develops a new framework for designing and analyzing convergent
finite difference methods for approximating both classical and viscosity
solutions of second order fully nonlinear partial differential equations (PDEs)
in 1-D. The goal of the paper is to extend the successful framework of
monotone, consistent, and stable finite difference methods for first order
fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs
such as Monge-Amp\`ere and Bellman type equations. New concepts of consistency,
generalized monotonicity, and stability are introduced; among them, the
generalized monotonicity and consistency, which are easier to verify in
practice, are natural extensions of the corresponding notions of finite
difference methods for first order fully nonlinear Hamilton-Jacobi equations.
The main component of the proposed framework is the concept of "numerical
operator", and the main idea used to design consistent, monotone and stable
finite difference methods is the concept of "numerical moment". These two new
concepts play the same roles as the "numerical Hamiltonian" and the "numerical
viscosity" play in the finite difference framework for first order fully
nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent
and monotone finite difference methods are proposed for second order fully
nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also
are proved to be stable and the second class contains Godunov-like methods.
Numerical results are also presented to gauge the performance of the proposed
finite difference methods and to validate the theoretical results of the paper.
