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

TL;DR

This paper develops a higher-order accurate geometric-volume conservation method for Cahn-Hilliard models, improving volume preservation in phase-field simulations through novel kernel design and integral-moment cancellation, validated by numerical benchmarks.

Contribution

It introduces a new kernel design and integral-moment cancellation condition to achieve third-order accuracy in volume conservation within Cahn-Hilliard models.

Findings

Achieves formal third-order accuracy in volume conservation.

Virtually eliminates artificial volume drift in simulations.

Prevents premature extinction of small droplets.

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 $\phi$ 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 $Q(\phi)$ 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.