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

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.

💡 Research Summary

The paper addresses a well‑known drawback of degenerate Cahn–Hilliard phase‑field models: when the interface thickness ε is finite, the geometric volume enclosed by the zero‑level set (the region where the order parameter ϕ is positive) drifts over time, even though the sharp‑interface limit conserves volume. Classical Cahn–Hilliard conserves the diffuse “mass” ∫Ωϕ dx, which approximates the geometric volume only to first order in ε. Zhou et al. (2022) proposed to replace mass conservation by exact conservation of a monotone mapping Q(ϕ) that more closely resembles a step function. The present work builds on that idea and makes three major contributions.

First, the authors perform a matched‑asymptotic expansion in the unscaled (physical) time variable, i.e. without the usual ε‑dependent time rescaling. In this formulation the normal velocity of the interface expands as V_n = ε V₁ + O(ε²); the leading term V₁ is the surface‑diffusion law V₁ = −∇_Γ·(M_Γ∇_Γ H). The inner coordinate z = d/ε (d is signed distance) yields the leading profile σ(z)=tanh(z/√2), and the first curvature‑induced correction Φ₁ satisfies a linear ODE LΦ₁ = √2/3 Q′(σ) − σ′, where L is self‑adjoint. The authors simplify the classical two‑integral representation of Φ₁ to a single quadrature in the phase variable u = σ(z): \