  In this paper we present a new first-order hyperbolic reformulation of the
Cahn-Hilliard equation. The model is obtained from the combination of augmented
Lagrangian techniques proposed earlier by the authors of this paper, with a
classical Cattaneo-type relaxation that allows to reformulate diffusion
equations as augmented first order hyperbolic systems with stiff relaxation
source terms. The proposed system is proven to be hyperbolic and to admit a
Lyapunov functional, in accordance with the original equations. A new numerical
scheme is proposed to solve the original Cahn-Hilliard equations based on
conservative semi-implicit finite differences, while the hyperbolic system was
numerically solved by means of a classical second order MUSCL-Hancock-type
finite volume scheme. The proposed approach is validated through a set of
classical benchmarks such as spinodal decomposition, Ostwald ripening and exact
stationary solutions.
