Our aim in this paper is to study a fully discrete scheme for modified higherorder
(in space) anisotropic generalized Cahn-Hilliard models which have extensive
applications in biology, image processing, etc. In particular, the scheme is a combination
of finite element or spectral method in space and a second-order stable scheme in
time. We obtain energy stability results, as well as the existence and uniqueness of the
numerical solution, both for the space semi-discrete and fully discrete cases. We also
give several numerical simulations which illustrate the theoretical results and, especially,
the effects of the higher-order terms on the anisotropy.