Third-Order Geometric-Volume Conservation in Cahn--Hilliard Models
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917241022316544 |
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| author | Musil, Josef |
| author_facet | Musil, Josef |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01497 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Third-Order Geometric-Volume Conservation in Cahn--Hilliard Models Musil, Josef Numerical Analysis Computational Physics Fluid Dynamics 65M08, 65M12, 76M12, 35K25 G.1.8; J.2 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. |
| title | Third-Order Geometric-Volume Conservation in Cahn--Hilliard Models |
| topic | Numerical Analysis Computational Physics Fluid Dynamics 65M08, 65M12, 76M12, 35K25 G.1.8; J.2 |
| url | https://arxiv.org/abs/2602.01497 |