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Diffuse interface treatment in generalized curvilinear coordinates with grid-adapting interface thickness

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proposes a parameter-free transformation of phase-field regularization into generalized curvilinear coordinates, making the diffuse interface thickness adapt to the local grid size while preserving Cartesian-order convergence.

desk verdict A genuinely new computational-space formulation with solid verification; fix the boundedness proof and address the interface-normal question before publication. read the letter →

arxiv 2411.18770 v1 pith:WMATH4X7 submitted 2024-11-27 physics.comp-ph

Henry Collis , Shahab Mirjalili , Ali Mani This is my paper · ORCID
classification physics.comp-ph
keywords phasefielddiffuseinterfacecurvilinearcoordinatesconservativeACDIgrid-adaptingthicknessboundednesstwo-phaseflow
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that phase-field two-phase methods, specifically the conservative diffuse interface (CDI) family and its ACDI variant, can be extended to generalized curvilinear grids without introducing a free parameter for interface thickness. The proposed move is to define the regularization flux in computational space, where the grid is uniform, and then project that flux back to physical space, so the interface automatically keeps the same number of cells across it everywhere. If true, this removes a known obstacle: direct transformations either make the interface too thick in refined regions or distort the interface through direction-dependent thickness parameters. Verification on drop advection, drop-in-shear, and a two-way coupled compressible Rayleigh-Taylor instability shows the same near-second-order convergence as on Cartesian grids, including on highly skewed grids. The practical payoff is that curvilinear and multi-block structured grids keep their efficiency advantages while phase-field interfaces remain stable, bounded, and grid-adaptive.

What carries the argument

The central object is the computational-space regularization-flux projection. Starting from the CDI flux written in computational coordinates, the physical flux is obtained by contraction with $\partial x_j/\partial \xi_i$, and the divergence is re-expressed in computational space via the metric Jacobian. The identity $(\partial x_j/\partial \xi_k)(\partial \xi_i/\partial x_j) = I$ collapses the projected equation to Eq. (10), where the sharpening normal uses computational-space gradients normalized by $|\nabla_\xi \phi|$. This is what makes the interface thickness constant in computational coordinates and therefore grid-adapting in physical space, and it also supplies the boundedness constraints $\epsilon_\xi/\Delta\xi \ge |U|_{\max}/\Gamma_\xi + 1/2$ and $\Delta t \le \Delta\xi^2/(2\Gamma_\xi \epsilon_\xi)$.

What would settle it

On a non-uniform, non-orthogonal grid with sharply varying Jacobian, initialize a stationary planar interface and measure the equilibrium profile thickness in physical space as a function of position; if the number of cells across the interface varies more than the local grid-size ratio, or if oscillations or values of $\phi$ outside $[0,1]$ appear for parameters satisfying the stated boundedness constraints, the central claim fails. Alternatively, run the drop-advection test on a grid whose mapping has discontinuous second derivatives and check whether the near-second-order convergence persists.

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Extended reading notes

Core claim

The paper claims that the ambiguity in curvilinear phase-field regularization—how to choose the interface-thickness parameter $\epsilon$ when the grid size varies—can be removed by formulating the regularization flux entirely in computational space and projecting it back to physical space. In the proposed form, $\epsilon_\xi$ is simply proportional to the computational grid spacing $\Delta\xi$ and $\Gamma_\xi$ to the maximum contravariant velocity, and the chain-rule projection reduces the final equation to the same shape as the Cartesian CDI operator with an additional $1/J$ factor. The resulting equilibrium interface has constant thickness in computational space, which projects to an interface that adapts to the local physical grid size. On the advection, shear-flow, and Rayleigh-Taylor tests, the curvilinear formulation converges at the same near-second-order rate as the Cartesian one on all grids tested, and the volume fraction remains bounded under the stated parameter and time-step constraints.

Load-bearing premise

The method assumes that the physically correct diffuse interface on a general grid is the one that has constant thickness in computational coordinates after the flux is projected back, an assumption verified here only on smooth, analytically defined grids.

Editorial extensions

If this is right

  • Phase-field parameters have a unique prescription on any curvilinear grid: $\epsilon_\xi \propto \Delta\xi$ and $\Gamma_\xi = \max_i |U_i|$, removing the option-1/option-2 ambiguity.
  • Interface thickness in physical space follows the local grid resolution, so refined regions get sharper interfaces without user intervention.
  • Boundedness of the volume fraction ($0 \le \phi \le 1$) carries over from Cartesian grids under the stated time-step and $\epsilon_\xi$ constraints for second-order centered discretizations.
  • The same derivation applies to CDI, ACDI, Allen-Cahn, and Cahn-Hilliard forms, as summarized in the paper's appendix.
  • On the tested flows, convergence is near second order and identical between Cartesian and curvilinear grids, with error magnitude growing with skewness but the convergence rate unchanged.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the transformation is derived purely from chain-rule projection, the same computational-space prescription could apply to other divergence-form regularization terms, including sharpening terms in level-set or volume-of-fluid contexts, though the paper does not test that.
  • The formulation naturally pairs with multi-block or locally refined structured grids, where each block carries its own $\Delta\xi$ and interface thickness would stay grid-adaptive across block boundaries; this is not demonstrated in the paper.
  • The constant-thickness-in-computational-space design implies that physical-space interface normals and curvatures derived directly from $\phi$ may reflect the computational-space profile rather than the physical one, so geometric postprocessing may need metric-corrected normals, which the paper leaves implicit.
  • A testable extension is to use the boundedness constraints to choose time steps adaptively in regions of strong grid stretching, since the relevant CFL condition depends on $\Gamma_\xi$ and $\Delta\xi$ rather than the physical velocity and grid size alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a curvilinear-coordinate formulation of conservative diffuse-interface (CDI/ACDI) phase-field equations. Instead of directly transforming the Cartesian regularization operator, the authors define the regularization in computational space with the standard thickness parameter ε_ξ = 0.75 Δξ and project the resulting flux back to physical space, yielding Eq. (10). They verify the formulation with three test problems — drop advection, drop in shear flow, and a two-way coupled compressible Rayleigh-Taylor instability — on four grids with varying skewness. The reported results show second-order L1 convergence for the advection and shear tests and visual grid independence for the Rayleigh-Taylor case.

Significance. If the claims hold, the paper provides a parameter-unambiguous extension of CDI/ACDI to structured curvilinear grids, with no fitted constants and a simple computational-space guideline for the interface thickness. The verification suite is appropriate in scope: three canonical cases, four grids, quantitative L1 convergence for two of them, and a genuinely coupled compressible problem. The paper also makes a credible effort to extend the boundedness argument from prior Cartesian work. The main weaknesses are that the physical-space meaning of the computational-space interface normal is not established, the Appendix B boundedness proof discretizes a different operator than Eq. (10) and a different advection scheme than the KEEP scheme used in Section 3, and the Rayleigh-Taylor grid-independence claim rests on visual contour overlays rather than a quantitative metric. These issues are local and fixable, but they are load-bearing for the central claims of accurate physical-space interface adaptation, boundedness without oscillations, and grid-independent convergence.

major comments (3)
  1. [2.2, Eq. (10)] The sharpening direction in physical space is not the physical interface normal on non-orthogonal grids. In Eq. (10) the nonlinear flux is proportional to ∂φ/∂ξ_i / |∇_ξ φ|; when interpreted as a physical vector this is (∂x_j/∂ξ_i)∂φ/∂ξ_i / |∇_ξ φ|, whereas the physical normal to the level sets is (∂ξ_i/∂x_j)∂φ/∂ξ_i / |∇_x φ|. These coincide only when the mapping is locally conformal, which the grids in Table 1 are not. Consequently the equilibrium profile is a constant-thickness tanh in computational coordinates, and in physical space its width and orientation depend on the local metric; it is not the usual isotropic diffuse interface. This is not necessarily a flaw if the method is intended as a computational-space regularization, but the paper should state that explicitly and justify it, because the abstract and Section 2.2 claim the transformation accurately adapts the interface thickness in physical space. The verification tests do not resolve the ambiguity: Eq. (14) initializes a circle in computational coordinates, so the advection and shear tests measure self-consistency on the same grid, and the Rayleigh-Taylor comparison in Fig. 12 is visual only. I recommend adding a quantitative test on a non-orthogonal grid with a flat interface whose physical normal is not aligned with a coordinate line, measuring the physical-space profile and normal, and reporting an interface-width error or a grid-to-grid difference metric for the RT case.
  2. [Appendix B, Eqs. (B.3)-(B.8)] The boundedness proof does not apply to the discretization actually used in the code. The diffusion term in Eq. (B.3) is written as Γ_ξ ε_ξ (φ_{i-1}/J_{i-1} - 2φ_i/J_i + φ_{i+1}/J_{i+1})/Δξ^2, which is a discretization of ∂^2(φ/J)/∂ξ^2, whereas Eq. (10) contains ∂/∂ξ_i [Γ_ξ(...)/J]; a consistent finite-volume discretization would evaluate 1/J at faces. The advection term is likewise a second-order central difference, not the sixth-order KEEP scheme described in Section 3. The proof therefore does not establish boundedness for the implemented scheme, and the statements in Section 2.2 and the Conclusion that boundedness has been extended to the curvilinear domain are stronger than what is shown. The authors already note that no formal proof exists for higher-order schemes, but the mismatch even at the second-order level should be corrected, or the boundedness claim should be downgraded to an empirical observation supported only by the test cases.
  3. [4.3, Fig. 12] The claim of grid-independent convergence for the Rayleigh-Taylor case is supported only by visual contour overlays. The text attributes the Grid 4 discrepancy to earlier wall interaction without a quantitative measure, so it is not possible to assess whether the four solutions converge to the same physical solution. Please provide a quantitative metric, for example the L1 or L2 difference of φ or of the interface position between grids at the common highest resolution, or a Richardson-type comparison of a scalar quantity such as the bubble tip position.
minor comments (4)
  1. [Table 1, Grid 3] The y-mapping for Grid 3 contains 'sin(λπξ1) sin(λπξ1)'; the second factor should probably be sin(λπξ2).
  2. [Section 4.1] The text says the radius is '0.25% of the computational domain width,' while Eq. (14) appears to use R = 0.25; please clarify which value is intended.
  3. [Eq. (13)] The definition of A_l uses the free index i without specifying the coordinate direction; since A_l is used as a flux component in Eq. (12), clarify whether A_l denotes a vector of components or the ξ1-component.
  4. [Section 3] The HTR solver and Legion framework are mentioned but no code or data availability statement is provided; adding one would strengthen the reproducibility of the verification results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the curvilinear CDI/ACDI formulation is derived from coordinate-transformation identities with no fitted constants, and the numerical studies are self-consistent verification rather than fitted predictions.

full rationale

The paper's core derivation (Eqs. 7-10) starts from the Cartesian CDI flux, re-expresses the regularization directly in computational coordinates, projects the flux back to physical space, and then takes a divergence. The simplification from Eq. 9 to Eq. 10 uses only the identity ∂xj/∂ξk ∂ξi/∂xj = I, so the final form is a coordinate transformation of the model, not a quantity fitted to the tests. The parameter choices ϵξ = 0.75Δξ and Γξ = max|Ui| follow the previously established boundedness criteria from [9,10]; these are cited parameter rules, not fitted outputs, and the paper reproduces the boundedness argument in Appendix B rather than merely invoking the citation. The numerical experiments (drop advection, drop in shear, Rayleigh-Taylor) compare against either the initial condition or a high-resolution solution on the same grids, which is standard verification of order of convergence and grid independence, not a hidden fit to the claimed result. The skeptical concern that the computational-space normal differs from the physical interface normal on non-orthogonal grids is a physical-correctness and validation issue, not a circularity: the paper does not derive that normal from the target结论, and the derivation of Eq. 10 is self-contained. Self-citations appear, but none is load-bearing for the central formulation, so the circularity score is 0.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The method's only hand-set parameter is the interface-thickness factor c_eps; the rest of the model comes from the prior CDI/ACDI literature and the four-equation compressible model. No new physical entities are introduced. The smooth-mapping and high-order-stability assumptions are the main unproved inputs.

free parameters (1)
  • interface thickness factor c_eps = eps_xi / Delta_xi = 0.75 (advection and RT); 0.7 stated in shear test without Delta_xi (possibly a typo for 0.75 Delta_xi)
    Chosen by hand following prior CDI practice to satisfy the boundedness condition Eq. B.1; not fitted to the convergence data. The shear-test value in Section 4.2 is ambiguous as printed.
assumptions (3)
  • domain assumption The four-equation thermal-mechanical equilibrium model with stiffened gas EOS is a valid representation for the two-phase flows considered.
    Invoked in Section 3 for the RT test; standard in the cited compressible multiphase literature [22-24,33-37].
  • domain assumption The coordinate transformation is smooth and non-degenerate with J > 0 everywhere.
    Required for the metric identities in Section 2 and for the positivity argument in Appendix B where J_i > 0 is assumed.
  • ad hoc to paper Boundedness constraints derived for second-order central schemes (Eqs. B.1, B.2) remain sufficient for the 6th-order KEEP advection scheme used in the tests.
    Stated in Appendix B: 'no formal proof currently exists... have held true for high-order schemes in practice'. The tests are the only evidence.

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Cite this review

Pith. "Pith review of Diffuse interface treatment in generalized curvilinear coordinates with grid-adapting interface thickness." pith.science (2026). https://pith.science/paper/WMATH4X7

@misc{pith2026241118770,
  author       = {Pith},
  title        = {Pith review of: Diffuse interface treatment in generalized curvilinear coordinates with grid-adapting interface thickness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMATH4X7}},
  note         = {Machine review of arXiv:2411.18770}
}
read the original abstract

A general approach for transforming phase field equations into generalized curvilinear coordinates is proposed in this work. The proposed transformation can be applied to isotropic, non-isotropic, and curvilinear grids without adding any ambiguity in determining the phase field parameters. Moreover, it accurately adapts the interface thickness to the local grid-size for a general curvilinear grid without creating oscillations. Three canonical verification tests are presented on four grids with varying skewness levels. The classic advection and drop in shear tests are extended to curvilinear grids and show that the original phase field on Cartesian grids and the proposed curvilinear form have an identical order of convergence. Additionally, the proposed method is shown to provide grid-independent convergence on a two-way coupled compressible Rayleigh-Taylor instability. These simulations illustrate the robustness and accuracy of the proposed method for handling complex interfacial structures on generalized curvilinear grids.

Figures

Figures reproduced from arXiv: 2411.18770 by the authors.

Figure 1
Figure 1. An example of a curvilinear domain in physical space is shown in the left panel and the corresponding computational [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The final shapes of a circular drop after 1 period of advection through a non-uniform rectilinear grid for two curvilinear [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Visual of a drop after one advection period for four different grids in computational space for the resolution ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Visual of a drop after one advection period for four different grids in physical space for the resolution ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Visual of the ϕ = 0.5 contour of a drop after one advection period for four different grids in computational space. Convergence of the drop is plotted for three varying resolutions. All contours are overlayed over a sample coarse grid with resolution (Nx, Ny) = (32, 32…
Figure 6
Figure 6. Figure 6: Visual convergence of the ϕ = 0.5 contour of a drop after one advection period for four different grids in physical space. Convergence of the drop is plotted using three varying resolutions. All contours are overlayed over a sample coarse grid with resolution (Nx, Ny) …
Figure 7
Figure 7. Figure 7: L1 error convergence of a drop after one advection period for four different grids at varying resolutions. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Visual of the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: L1 error convergence of drop in shear flow on curvilinear grids of varying skewness. [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Raleigh Taylor simulation on four different grids. Visual convergence of the [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Visual convergence of the ϕ = 0.5 contour of a Rayleigh-Taylor instability at t = 1.0 for four different grids in computational space. Convergence of the contour is plotted using five varying resolutions. All contours are overlayed over a sample coarse grid with resol…
Figure 12
Figure 12. Figure 12: Visual convergence of the ϕ = 0.5 contour of the center Rayleigh-Taylor instability at t = 1.0 between four different curvilinear grids at the resolution (Nx, Ny) = (1600, 1600). proofs for ACDI can be carried out as well). The proposed form of the ACDI curvilinear ph…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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