Paper · Jun 6, 2025

Elementary discrete diffusion/redistancing schemes for the mean curvature flow

Antonin Chambolle, Daniele De Gennaro, Massimiliano Morini

math.APWider flowsarXiv:2506.05946v2
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Abstract

We consider a fully discrete and explicit scheme for the mean curvature flow of boundaries, based on an elementary diffusion step and a precise redistancing operation. We give an elementary convergence proof for the scheme under the standard CFL condition , where is the time discretization step and the space step. We discuss extensions to more general convolution/redistancing schemes.

Mean curvature flow

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions (Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al., 2021)5Julin 2021 Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus (Daniele De Gennaro and Anna Kubin, 2022)4De Gennaro 2022 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)2Chambolle 2022 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)3De Gennaro 2023 Discrete-to-continuum crystalline curvature flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2024)1Chambolle 2024 Median filter method for mean curvature flow using a random Jacobi algorithm (Anton Ullrich and Tim Laux, 2024)8Ullrich 2024 A distributional approach to nonlocal curvature flows (Filippo Cagnetti, Massimiliano Morini and Dario Reggiani, 2025)6Cagnetti 2025 Variational Nonlinear and Nonlocal Curvature Flows (Daniele De Gennaro, 2025)7De Gennaro 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Mean curvature flow · Same authors Antonin Chambolle, Daniele De Gennaro, Massimiliano Morini · Shared terms “fully discrete”, “discretization”

  2. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Mean curvature flow · Same authors Antonin Chambolle, Daniele De Gennaro, Massimiliano Morini

  3. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows PDF

    Topics Mean curvature flow · Same author Daniele De Gennaro · Shared terms “diffusion”

  4. Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

    Daniele De Gennaro and Anna KubinWider flows PDF

    Topics Mean curvature flow · Same author Daniele De Gennaro

  5. The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

    Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al.Wider flows PDF

    Topics Mean curvature flow · Same author Massimiliano Morini

  6. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows PDF

    Topics Mean curvature flow · Same author Massimiliano Morini

  7. Variational Nonlinear and Nonlocal Curvature Flows

    Daniele De GennaroWider flows PDF

    Same author Daniele De Gennaro

  8. Median filter method for mean curvature flow using a random Jacobi algorithm

    Anton Ullrich and Tim LauxWider flows PDF

    Topics Mean curvature flow · Shared terms “discretization”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.