  Comparing with the classical local gradient flow and phase field models, the
nonlocal models such as nonlocal Cahn-Hilliard equations equipped with nonlocal
diffusion operator can describe more practical phenomena for modeling phase
transitions. In this paper, we construct an accurate and efficient scalar
auxiliary variable approach for the nonlocal Cahn-Hilliard equation with
general nonlinear potential. The first contribution is that we have proved the
unconditional energy stability for nonlocal Cahn-Hilliard model and its
semi-discrete schemes carefully and rigorously. Secondly, what we need to focus
on is that the non-locality of the nonlocal diffusion term will lead the
stiffness matrix to be almost full matrix which generates huge computational
work and memory requirement. For spatial discretizaion by finite difference
method, we find that the discretizaition for nonlocal operator will lead to a
block-Toeplitz-Toeplitz-block (BTTB) matrix by applying four transformation
operators. Based on this special structure, we present a fast procedure to
reduce the computational work and memory requirement. Finally, several
numerical simulations are demonstrated to verify the accuracy and efficiency of
our proposed schemes.
