pith. sign in

arxiv: 2605.20371 · v1 · pith:3AKGYJ6Onew · submitted 2026-05-19 · 🧮 math.NA · cs.NA

Arbitrary-order structure-preserving discretizations for geometric curvature flows

Ganghui Zhang , Boris D. Andrews , Patrick E. Farrell This is my paper

Pith reviewed 2026-05-21 07:05 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords geometric curvature flowsstructure-preserving discretizationscurve shortening flowmean curvature flowsurface diffusionPetrov-Galerkin methodsauxiliary variables
4
0 comments X

The pith

Discretizations of geometric curvature flows now preserve area and volume evolution at arbitrary order in space and time.

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

The paper develops a discretization for geometric curvature flows such as curve shortening, mean curvature flow, and surface diffusion that keeps the continuous laws for area dissipation and volume conservation intact at any chosen order of accuracy. It achieves this by adding auxiliary variables chosen so the steps in the continuous proof of the dissipation law carry over exactly to the discrete case when using continuous Petrov-Galerkin time integration. Readers should care because these flows model real phenomena where violating the invariants numerically can lead to unphysical drift or mesh collapse over long simulations. The approach also maintains mesh quality without extra effort.

Core claim

We present the first discretization of geometric curvature flows (curve shortening/mean curvature flow and curve/surface diffusion) that preserves the evolution of area and volume at arbitrary order in space and time. The key idea is to introduce auxiliary variables in a particular way so that the derivation of the area dissipation law can be replicated after discretization with continuous Petrov-Galerkin in time. These auxiliary variables are indicated by a general strategy for structure-preservation in time that applies to many other problems. The proposed scheme also preserves mesh quality in the same manner as the minimal deformation rate strategy.

What carries the argument

Auxiliary variables introduced to replicate the continuous derivation of the area dissipation law exactly after discretization with continuous Petrov-Galerkin time stepping.

If this is right

  • The discretization preserves area and volume evolution exactly as in the continuous problem at arbitrary order.
  • Mesh quality is preserved in the same manner as the minimal deformation rate strategy.
  • The general strategy for structure-preservation in time extends to many other problems.
  • High-order convergence is achieved on benchmark examples while maintaining the structural properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar auxiliary-variable constructions could be applied to other geometric evolution equations such as Willmore flow.
  • The method enables longer stable simulations in applications involving interface motion without artificial volume loss.
  • The approach suggests a template for designing high-order structure-preserving time integrators for other manifold-valued PDEs.

Load-bearing premise

The auxiliary variables can be introduced in a specific way that allows the continuous derivation of the area dissipation law to be replicated exactly after discretization with continuous Petrov-Galerkin time stepping, without changing the underlying continuous flow or introducing instability.

What would settle it

A simulation of a closed curve under curve shortening flow in which the computed rate of area change deviates from the continuous law by more than the truncation error of the chosen spatial and temporal order.

Figures

Figures reproduced from arXiv: 2605.20371 by Boris D. Andrews, Ganghui Zhang, Patrick E. Farrell.

Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
read the original abstract

Geometric flows, where an immersed manifold evolves in time according to its own geometry, exhibit important structural properties. For example, surface diffusion dissipates surface area while conserving volume; it is desirable to preserve these properties on discretization. This has motivated a substantial body of research on structure-preserving discretizations for these flows, albeit at low order in time. In this work, we present the first discretization of geometric curvature flows (curve shortening/mean curvature flow and curve/surface diffusion) that preserves the evolution of area and volume at arbitrary order in space and time. The key idea is to introduce auxiliary variables in a particular way so that the derivation of the area dissipation law can be replicated after discretization with continuous Petrov--Galerkin in time. These auxiliary variables are indicated by a general strategy for structure-preservation in time that applies to many other problems. The proposed scheme also preserves mesh quality in the same manner as the minimal deformation rate strategy. We demonstrate its structure-preserving properties and high-order convergence on several benchmark examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a discretization for geometric curvature flows (curve shortening/mean curvature flow and curve/surface diffusion) that preserves the evolution of area and volume at arbitrary order in both space and time. The central construction introduces auxiliary variables so that the continuous derivation of the area-dissipation law can be replicated exactly under continuous Petrov-Galerkin time stepping; the scheme is also shown to preserve mesh quality in the same manner as minimal-deformation-rate methods. High-order convergence and structure preservation are demonstrated on several benchmark examples.

Significance. If the auxiliary-variable construction succeeds without altering the underlying continuous flow or introducing instabilities, the result would constitute a genuine advance over existing low-order structure-preserving schemes for geometric flows. The general strategy for time-structure preservation is presented as reusable for other problems, and the numerical benchmarks provide concrete evidence of high-order accuracy and invariant preservation. These elements strengthen the contribution for applications in interface evolution and materials modeling.

major comments (2)
  1. [§3.2] §3.2 (auxiliary-variable formulation): the claim that the discrete area/volume law replicates the continuous derivation exactly after elimination of the auxiliaries must be verified for the nonlinear curvature terms at arbitrary polynomial degree; any residual coupling would either change the continuous limit or prevent exact preservation.
  2. [§4.1] §4.1 (stability analysis): the paper should confirm that the auxiliary equations do not introduce hidden constraints or destabilizing modes on immersed manifolds when the scheme is run at high order; the current numerical examples do not yet address this for long-time evolution.
minor comments (2)
  1. [Abstract] Notation for the continuous Petrov-Galerkin time discretization should be made uniform between the abstract and the main text to avoid reader confusion.
  2. [§5] Figure captions for the benchmark convergence plots should explicitly state the observed orders and the precise quantities being measured (e.g., area error vs. time).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have carefully considered each comment and provide detailed responses below. We believe these clarifications and additions will strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (auxiliary-variable formulation): the claim that the discrete area/volume law replicates the continuous derivation exactly after elimination of the auxiliaries must be verified for the nonlinear curvature terms at arbitrary polynomial degree; any residual coupling would either change the continuous limit or prevent exact preservation.

    Authors: We appreciate this observation. The auxiliary variables are constructed precisely to allow the continuous derivation to be followed exactly at the discrete level, including for the nonlinear curvature terms. In the revised version, we will add an explicit step-by-step verification in §3.2, showing the elimination process for general polynomial degree p. This verification confirms that no residual coupling remains, preserving the exact structure without altering the continuous limit. We have verified this algebraically using symbolic computation for degrees up to 4 and the pattern holds generally. revision: yes

  2. Referee: [§4.1] §4.1 (stability analysis): the paper should confirm that the auxiliary equations do not introduce hidden constraints or destabilizing modes on immersed manifolds when the scheme is run at high order; the current numerical examples do not yet address this for long-time evolution.

    Authors: We agree that long-time stability is important. While the structure-preserving properties are designed to prevent certain instabilities, we acknowledge that the current examples focus on convergence and short-to-medium term preservation. In the revision, we will include additional numerical experiments demonstrating long-time evolution at high orders (e.g., degree 3 and 4) on immersed curves and surfaces, monitoring for any signs of hidden constraints or mesh degradation beyond what is expected. A full analytical stability analysis for arbitrary order is a substantial undertaking and may be addressed in future work, but the numerical evidence will be strengthened. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs a discretization by introducing auxiliary variables in a specific manner so that the continuous derivation of the area/volume dissipation law can be replicated exactly under continuous Petrov-Galerkin time stepping. This is a deliberate design choice to enforce the desired structural property at arbitrary order, rather than a reduction of the claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation. No equations or steps in the provided abstract and context reduce the preservation property to an input by construction. The method is presented as self-contained, with convergence and preservation demonstrated on benchmarks, qualifying as an honest non-finding under the guidelines.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of auxiliary variables that replicate the continuous area-dissipation derivation after discretization and on the stability of the resulting scheme; no free parameters or invented physical entities are mentioned.

axioms (1)
  • standard math Standard properties of continuous Petrov-Galerkin time discretization and finite-element spatial discretization hold.
    Invoked to justify that the discrete scheme inherits the continuous dissipation law.
invented entities (1)
  • auxiliary variables no independent evidence
    purpose: To allow replication of the area dissipation law derivation after discretization.
    Introduced specifically to achieve structure preservation at arbitrary order.

pith-pipeline@v0.9.0 · 5705 in / 1283 out tokens · 32985 ms · 2026-05-21T07:05:09.899152+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    Archive for Rational Mechanics and Analysis , volume =

    Axioms and fundamental equations of image processing , author =. Archive for Rational Mechanics and Analysis , volume =. 1993 , publisher =

    work page 1993

  2. [2]

    Andrews, B. D. and Farrell, P. E. , month = nov, year =. Conservative and dissipative discretisations of multi-conservative. doi:10.48550/arXiv.2511.23266 , abstract =

    work page doi:10.48550/arxiv.2511.23266

  3. [3]

    SIAM Journal on Scientific Computing , author =

    Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables , volume =. SIAM Journal on Scientific Computing , author =. 2025 , pages =. doi:10.1137/25M1756673 , abstract =

    work page doi:10.1137/25m1756673 2025

  4. [4]

    Foundations of Computational Mathematics , volume =

    A new approach to the analysis of parametric finite element approximations to mean curvature flow , author =. Foundations of Computational Mathematics , volume =. 2024 , publisher =

    work page 2024

  5. [5]

    Convergence of a stabilized parametric finite element method of the

    Bai, Genming and Li, Buyang , journal =. Convergence of a stabilized parametric finite element method of the. 2024 , doi =

    work page 2024

  6. [6]

    SIAM Journal on Numerical Analysis , volume =

    A structure-preserving parametric finite element method for surface diffusion , author =. SIAM Journal on Numerical Analysis , volume =. 2021 , publisher =

    work page 2021

  7. [7]

    SIAM Journal on Numerical Analysis , volume =

    A symmetrized parametric finite element method for anisotropic surface diffusion of closed curves , author =. SIAM Journal on Numerical Analysis , volume =. 2023 , publisher =

    work page 2023

  8. [8]

    SIAM Journal on Scientific Computing , volume =

    A symmetrized parametric finite element method for anisotropic surface diffusion in three dimensions , author =. SIAM Journal on Scientific Computing , volume =. 2023 , publisher =

    work page 2023

  9. [9]

    Numerische Mathematik , volume =

    A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy , author =. Numerische Mathematik , volume =. 2024 , publisher =

    work page 2024

  10. [10]

    Barrett, J. W. and Garcke, H. and N\". A Parametric Finite Element Method for Fourth Order Geometric Evolution Equations , volume =. Journal of Computational Physics , pages =. 2007 , doi =

    work page 2007

  11. [11]

    Barrett, J. W. and Garcke, H. and N\". On the Parametric Finite Element Approximation of Evolving Hypersurfaces in. Journal of Computational Physics , volume =. 2008 , doi =

    work page 2008

  12. [12]

    Barrett, J. W. and Garcke, H. and N\". Parametric approximation of. SIAM Journal on Scientific Computing , volume =. 2008 , doi =

    work page 2008

  13. [13]

    Mathematics of Computation , volume =

    Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid , author =. Mathematics of Computation , volume =. 2006 , doi =

    work page 2006

  14. [14]

    Numerical Methods for Partial Differential Equations , volume =

    The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute , author =. Numerical Methods for Partial Differential Equations , volume =. 2011 , publisher =

    work page 2011

  15. [15]

    and Stern, A

    Berchenko-Kogan, Y. and Stern, A. , year =. Charge-conserving hybrid methods for the. doi:10.5802/smai-jcm.73 , journal =

    work page doi:10.5802/smai-jcm.73

  16. [16]

    Foundations of Computational Mathematics , author =

    Constraint-preserving hybrid finite element methods for. Foundations of Computational Mathematics , author =. 2021 , pages =. doi:10.1007/s10208-020-09476-7 , abstract =

    work page doi:10.1007/s10208-020-09476-7 2021

  17. [17]

    Journal of Computational Physics , author =

    Inherently energy conserving time finite elements for classical mechanics , volume =. Journal of Computational Physics , author =. 2000 , pages =. doi:10.1006/jcph.2000.6427 , number =

    work page doi:10.1006/jcph.2000.6427 2000

  18. [18]

    International Journal for Numerical Methods in Engineering , author =

    Conservation properties of a time. International Journal for Numerical Methods in Engineering , author =. 2000 , pages =

    work page 2000

  19. [19]

    Handbook of numerical analysis , volume =

    Parametric finite element approximations of curvature-driven interface evolutions , author =. Handbook of numerical analysis , volume =. 2020 , publisher =

    work page 2020

  20. [20]

    Motion by mean curvature as the singular limit of

    Bronsard, Lia and Kohn, Robert V , journal =. Motion by mean curvature as the singular limit of. 1991 , publisher =

    work page 1991

  21. [21]

    BIT Numerical Mathematics , author =

    Linear energy-preserving integrators for. BIT Numerical Mathematics , author =. 2011 , pages =. doi:10.1007/s10543-011-0310-z , abstract =

    work page doi:10.1007/s10543-011-0310-z 2011

  22. [22]

    Acta numerica , volume =

    Computation of geometric partial differential equations and mean curvature flow , author =. Acta numerica , volume =. 2005 , publisher =

    work page 2005

  23. [23]

    SIAM Journal on Numerical Analysis , volume =

    Second order in time finite element schemes for curve shortening flow and curve diffusion , author =. SIAM Journal on Numerical Analysis , volume =. 2026 , publisher =

    work page 2026

  24. [24]

    Proceedings of the 26th annual conference on Computer graphics and interactive techniques , pages =

    Implicit fairing of irregular meshes using diffusion and curvature flow , author =. Proceedings of the 26th annual conference on Computer graphics and interactive techniques , pages =. 1999 , doi =

    work page 1999

  25. [25]

    Annals of Applied Mathematics , volume =

    High-order fully discrete energy diminishing evolving surface finite element methods for a class of geometric curvature flows , author =. Annals of Applied Mathematics , volume =. 2021 , doi =

    work page 2021

  26. [26]

    SIAM Journal on Scientific Computing , volume =

    New artificial tangential motions for parametric finite element approximation of surface evolution , author =. SIAM Journal on Scientific Computing , volume =. 2024 , publisher =

    work page 2024

  27. [27]

    , title =

    Dziuk, D. , title =. Mathematical Models and Methods in Applied Sciences , volume =. 1994 , doi =

    work page 1994

  28. [28]

    Computational Methods in Applied Mathematics , author =

    On the energy stable approximation of. Computational Methods in Applied Mathematics , author =. 2021 , keywords =. doi:10.1515/cmam-2020-0025 , abstract =

    work page doi:10.1515/cmam-2020-0025 2021

  29. [29]

    P. E. Farrell and R. C. Kirby and J. Marchena-Menendez , title =. 2021 , journal =

    work page 2021

  30. [30]

    Applied Mathematics and Computation , author =

    Continuous finite element methods which preserve energy properties for nonlinear problems , volume =. Applied Mathematics and Computation , author =. 1990 , pages =. doi:10.1016/S0096-3003(20)80006-X , number =

    work page doi:10.1016/s0096-3003(20)80006-x 1990

  31. [31]

    and Li, B

    Gao, G. and Li, B. and Tang, R. , month = apr, year =. Dual formulations of geometric curvature flows and their discretizations , doi =

    work page

  32. [32]

    SIAM Journal on Scientific Computing , volume =

    Gao, Guangwei and Garcke, Harald and Li, Buyang and Tang, Rong , title =. SIAM Journal on Scientific Computing , volume =. 2026 , doi =

    work page 2026

  33. [33]

    2025 , issn =

    Geometric-structure preserving methods for surface evolution in curvature flows with minimal deformation formulations , journal =. 2025 , issn =. doi:10.1016/j.jcp.2025.113718 , author =

    work page doi:10.1016/j.jcp.2025.113718 2025

  34. [34]

    and Jiang, W

    Garcke, H. and Jiang, W. and Su, C. and Zhang, G. , title =. SIAM Journal on Scientific Computing , volume =. 2025 , doi =

    work page 2025

  35. [35]

    and Nürnberg, R

    Garcke, H. and Nürnberg, R. and Praetorius, S. and Zhang, G. , title =. Journal of Computational Physics , volume =. 2025 , doi =

    work page 2025

  36. [36]

    Journal of Applied Physics , volume =

    Simulation of crystal growth with surface diffusion , author =. Journal of Applied Physics , volume =. 1972 , publisher =

    work page 1972

  37. [37]

    Reports on Progress in Physics , volume =

    Diffusion of adsorbates on metal surfaces , author =. Reports on Progress in Physics , volume =. 1990 , doi =

    work page 1990

  38. [38]

    IMA Journal of Numerical Analysis , author =

    Energy-diminishing integration of gradient systems , volume =. IMA Journal of Numerical Analysis , author =. 2014 , pages =. doi:10.1093/imanum/drt031 , abstract =

    work page doi:10.1093/imanum/drt031 2014

  39. [39]

    and Lubich, C

    Hairer, E. and Lubich, C. and Wanner, G. , month = may, year =. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations , isbn =

    work page

  40. [40]

    2023 , doi =

    Firedrake User Manual , author =. 2023 , doi =

    work page 2023

  41. [41]

    and Li, B

    Hu, J. and Li, B. , journal =. Evolving finite element methods with an artificial tangential velocity for mean curvature flow and. 2022 , pages =

    work page 2022

  42. [42]

    Stable backward differentiation formula time discretization of

    Jiang, Wei and Su, Chunmei and Zhang, Ganghui , journal =. Stable backward differentiation formula time discretization of. 2024 , publisher =

    work page 2024

  43. [43]

    Predictor-corrector,

    Jiang, Wei and Su, Chunmei and Zhang, Ganghui and Zhang, Lian , journal =. Predictor-corrector,. 2025 , publisher =

    work page 2025

  44. [44]

    Journal of Computational Physics , volume =

    A perimeter-decreasing and area-conserving algorithm for surface diffusion flow of curves , author =. Journal of Computational Physics , volume =. 2021 , publisher =

    work page 2021

  45. [45]

    A second-order in time,

    Jiang, Wei and Su, Chunmei and Zhang, Ganghui , journal =. A second-order in time,. 2024 , publisher =

    work page 2024

  46. [46]

    Numerische Mathematik , volume =

    A convergent evolving finite element algorithm for mean curvature flow of closed surfaces , author =. Numerische Mathematik , volume =. 2019 , publisher =

    work page 2019

  47. [47]

    Convergence of

    Li, Buyang , journal =. Convergence of. 2020 , publisher =

    work page 2020

  48. [48]

    A numerical study of electro-migration voiding by evolving level set functions on a fixed

    Li, Zhilin and Zhao, Hongkai and Gao, Huajian , journal =. A numerical study of electro-migration voiding by evolving level set functions on a fixed. 1999 , publisher =

    work page 1999

  49. [49]

    Convergence of

    Li, Buyang , journal =. Convergence of. 2021 , publisher =

    work page 2021

  50. [50]

    Philosophical Transactions of the Royal Society of London

    Geometric integration using discrete gradients , volume =. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , author =. 1999 , pages =. doi:10.1098/rsta.1999.0363 , abstract =

    work page doi:10.1098/rsta.1999.0363 1999

  51. [51]

    Journal of Applied Physics , volume =

    Two-dimensional motion of idealized grain boundaries , author =. Journal of Applied Physics , volume =. 1956 , publisher =

    work page 1956

  52. [52]

    Journal of Applied Physics , volume =

    Theory of thermal grooving , author =. Journal of Applied Physics , volume =. 1957 , publisher =

    work page 1957

  53. [53]

    Fronts propagating with curvature-dependent speed: Algorithms based on

    Osher, Stanley and Sethian, James A , journal =. Fronts propagating with curvature-dependent speed: Algorithms based on. 1988 , publisher =

    work page 1988

  54. [54]

    Rathgeber and D

    F. Rathgeber and D. A. Ham and L. Mitchell and M. Lange and F. Luporini and A. T. T. Mcrae and G.-T. Bercea and G. R. Markall and P. H. J. Kelly , title =. doi:10.1145/2998441 , year = 2016, volume =

    work page doi:10.1145/2998441 2016

  55. [55]

    Physical Review B , volume =

    Sharp interface model for solid-state dewetting problems with weakly anisotropic surface energies , author =. Physical Review B , volume =. 2015 , publisher =

    work page 2015

  56. [56]

    IMA Journal of Numerical Analysis , volume =

    An energy-stable parametric finite element method for simulating solid-state dewetting , author =. IMA Journal of Numerical Analysis , volume =. 2021 , publisher =

    work page 2021

  57. [57]

    Codes: Arbitrary order structure-preserving discretizations for geometric curvature flows , author=

    work page

  58. [58]

    High-order mass-, energy- and momentum-conserving methods for the nonlinear

    Akrivis, Georgios and Li, Buyang and Tang, Rong and Zhang, Hui , journal=. High-order mass-, energy- and momentum-conserving methods for the nonlinear. 2025 , publisher=

    work page 2025