A first-order hyperbolic reformulation of the Cahn-Hilliard equation
<|reference_start|>A first-order hyperbolic reformulation of the Cahn-Hilliard equation: 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.<|reference_end|>
@article{dhaouadi2024a,
    title={A first-order hyperbolic reformulation of the Cahn-Hilliard equation},
    author={Firas Dhaouadi, Michael Dumbser, Sergey Gavrilyuk},
    journal={arXiv preprint arXiv:2408.03862},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.03862},
    primaryClass={math.NA cs.NA math.AP}
}