Third-Order Geometric-Volume Conservation in Cahn--Hilliard Models

TL;DR

The paper addresses artificial geometric-volume drift in diffuse-interface Cahn--Hilliard models with finite interface thickness $\varepsilon$ and proposes a conservation-improved CH--IC framework that exactly conserves a designed monotone mapping $Q(\phi)$. By performing unscaled matched asymptotics, deriving the first curvature-corrected inner profile $\Phi_1$, and identifying a moment-cancellation condition $\mathcal{C}_1[Q]=\mathcal{M}_1[Q]+\mathcal{J}_1[Q]=0$, the authors enable a practical inverse-design workflow to achieve formal third-order accuracy in geometric-volume conservation with respect to $\varepsilon$. Kernel families, including exponential and Padé-type enrichments, are tuned within low-dimensional parameter spaces to satisfy the balance condition while maintaining endpoint degeneracy, yielding $\mathrm{Err}_{\mathcal{V}}(t)=\mathcal{O}(\varepsilon^3)$. The proposed unconditional energy-dissipative discretization preserves the discrete conserved quantity, and numerical experiments on multi-scale droplet coarsening and high-curvature geometry demonstrate substantial reduction of artificial drift and improved droplet longevity with only modest computational overhead. The work provides a robust blueprint for high-fidelity CH-based simulations and informs future coupling with hydrodynamics or other physics via preserved volume and gradient-flow structure.

Abstract

Degenerate Cahn-Hilliard phase-field models provide a robust approximation of surface-diffusion-driven interface motion without explicit front tracking. In computations, however, the geometric volume enclosed by the interface -- the region where the order parameter $φ$ is positive -- may drift at finite interface thickness, producing artificial shrinkage or growth even when the sharp-interface limit conserves volume. We revisit and extend the improved-conservation framework of Zhou et al., where one replaces classical mass conservation by the exact conservation of a designed monotone mapping $Q(φ)$ that more accurately approximates a step function. Building on this framework, we (i) carry out the matched-asymptotic analysis in the unscaled physical time formulation, (ii) derive a simplified representation of the first-order inner correction to the interface profile, and (iii) identify an integral-moment cancellation condition that controls the leading geometric-volume defect. This mechanism becomes a practical design rule: we select regularization kernels within parameterized families -- including exponential and Pade-type -- to reach effective higher-order behavior and satisfy the cancellation condition at moderate parameter values. As a result, the proposed kernels achieve formal third-order accuracy in geometric-volume conservation with respect to interface thickness. Finally, we describe an unconditional energy-dissipative numerical discretization that exactly preserves the discrete conserved quantity. Numerical benchmarks on multi-scale droplet coarsening and shape relaxation demonstrate that the moment-balanced kernels virtually eliminate artificial drift and prevent premature extinction of small droplets.

Figures (14)

• Figure 1: Geometry and notation for the sharp interface limit.
• Figure 2: Conservation mapping $Q_k(\phi)$ and its derivative $Q_k'(\phi)$ for Zhou's polynomial family \ref{['eq:Qk_def']} with degeneracy orders $k = 1,2,3,8$. (a) The mapping $Q_k$ increasingly approximates the sharp sign function $\operatorname{sgn}(\phi)$ (dotted grey) as $k$ grows. (b) The derivative $Q_k'$ concentrates near $\phi = 0$ and vanishes at $\phi = \pm 1$ to order $k$, controlling the endpoint degeneracy. The case $k = 1$ corresponds to the NMN model of Bretin et al. bretin_2022.
• Figure 3: Comparison of kernel families. (a) Conserved mapping $Q(\phi)$. (b) Kernel $Q'(\phi)$: moment-balanced designs concentrate near $\phi=0$. (c) First inner correction $\Phi_1(z)/H$: NMN (solid) has $\Phi_1\equiv 0$; others develop curvature-induced distortion. (d) Combined moment $\mathcal{C}_1$: five balanced designs achieve $\mathcal{C}_1\approx0$.
• Figure 4: Initial condition for the multi-scale coarsening study: four droplets with radii $R \in \{0.15, 0.10, 0.06, 0.03\}$ on a $4\times 1$ domain.
• Figure 5: Volume conservation error $\mathrm{Err}_{\mathcal{V}}(t)$ during coarsening.

Theorems & Definitions (24)

• Remark 1: Mobility magnitude and effective time scale
• Lemma 1
• Remark 2: Advection and coupled transport
• Remark 3: Normalization matters for shaped kernels
• Remark 4: Interpretation
• Theorem 1: Single-integral formula for $\mathcal{M}_1$
• proof
• Lemma 2: Strict negativity of $\mathcal{M}_1$
• proof
• Remark 5: Closed forms for $S\equiv 1$ and why shaped kernels are different