A first-order hyperbolic reformulation of the Cahn-Hilliard equation

TL;DR

This paper introduces a novel first-order hyperbolic reformulation of the Cahn-Hilliard equation, enabling new numerical approaches and analysis while preserving key properties of the original model.

Contribution

It presents a new hyperbolic reformulation combining augmented Lagrangian and Cattaneo-type relaxation, with proven hyperbolicity and a Lyapunov functional.

Findings

The reformulated system is hyperbolic and stable.

Numerical schemes effectively solve the reformulated and original equations.

Validation on classical benchmarks confirms the approach's accuracy.

Abstract

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.