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

TL;DR

The paper develops a first-order hyperbolic reformulation of the Cahn-Hilliard equation by combining an augmented-Lagrangian approach with Maxwell–Cattaneo-type relaxation, yielding a hyperbolic, mass-conserving system that admits a Lyapunov functional. A companion numerical strategy uses a conservative semi-implicit finite-difference scheme for the original CH equation and a second-order MUSCL-Hancock finite-volume scheme for the hyperbolic reformulation, enabling efficient and stable simulations. Formal asymptotic analysis shows the hyperbolic system is consistent with CH for small $\gamma$ under scalings $\alpha=\gamma^{-1}$, $\tau=\gamma^2$, $\beta=\gamma^2$, bridging the two formulations. Extensive 1D and 2D benchmarks (spinodal decomposition, Ostwald ripening, exact stationary solutions, and radially symmetric equilibria) demonstrate excellent agreement with CH1, maintaining mass conservation and bounded energy dissipation, and highlighting the approach's practicality for stiff, high-order phase-field problems.

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.

Figures (9)

• Figure 1: Comparison of a stationary solution of the hyperbolic Cahn-Hilliard model (discontinuous lines) with the original counterpart (solid line) for different values of the penalty parameter $\alpha$. The displayed solutions are obtained by solving the initial-value problems \ref{['eq:ODE1']} and \ref{['eq:ODE2']} on the domain $x\in[0,0.6]$ discretized with a constant step of size $\Delta x=10^{-5}$.
• Figure 2: Comparison of the numerical solution obtained by solving numerically \ref{['eq:CH_Hyp']} (shaped dots), with the equivalent exact solution of \ref{['eq:CH1']}, which is provided as initial data (black line). The left graphic compares $c(x)$ (red squares) and $\varphi(x)$ (green circles) to the initial data. The right graphic compares the numerically evaluated $\partial c/\partial x$ (red squares) and $p(x)$ (green circles) to the gradient of the exact solution.
• Figure 3: Left: Relative error of the total energy over time for several values of $\tau$ and a fixed mesh resolution of $N=2000$. Center: Relative error of the total energy over time for a fixed value of $\tau = 10^{-4}$ and several mesh resolutions. The dashed lines correspond to the solution obtained by numerically integrating the ode \ref{['eq:Edecay_int']}. Right: Numerical integral of $c$ over time for several mesh sizes (discrete mass conservation).
• Figure 4: Comparison of the numerical results for the spinodal decomposition test case between the original Cahn-Hilliard model (orange dash-dotted lined) and its hyperbolic counterpart \ref{['eq:CH_Hyp']} (solid black line), for several times.
• Figure 5: Left: Comparison of the numerical results for the initial conditions $IC^{1-3}$ (green triangles, blue squares and yellow squares, respectively) with the numerical simulation obtained with a well-prepared initial condition $c^{wp}$ (red circles) and $\hat{c}$ (solid black line). Right: Comparison of $p$ and $c_x$ for the numerical solutions $c^1$ and $c^{wp}$. Comparison is done at $t=0.14$ for the same parameters as \ref{['fig:spinodal']}.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Proposition
• proof