Consistency of a large time-step scheme for mean curvature motion

  • M. Falcone
  • R. Ferretti
Conference paper


We propose a new scheme for the level set approximation of motion by mean curvature (MCM). The scheme originates from a representation formula recently given by Soner and Touzi, which allows us to construct large time-step, Godunov-type schemes. One such scheme is presented and its consistency is analyzed. We also provide and discuss some numerical tests.


Viscosity Solution Representation Formula Level Curve Finite Difference Approximation Local Truncation Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barles, G., Georgelin, Ch. (1995): A simple proof of the convergence of an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32, 484–500MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Chen, Y.-G., Giga, Y., Goto, S. (1991): Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33, 749–786MathSciNetMATHGoogle Scholar
  3. [3]
    Evans, L.C. (1993): Convergence of an algorithm for mean curvature motion. Indiana Univ. Math. J. 42, 533–557MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Evans, L.C. (1997): Regularity for fully nonlinear elliptic equations and motion by mean curvature. In: Bardi, M. et al. (eds.): Viscosity solutions and applications (Lecture Notes in Mathematics, vol. 1660). Springer, Berlin, pp. 98–133CrossRefGoogle Scholar
  5. [5]
    Evans, L.C, Spruck, J. (1991): Motion of level sets by mean curvature. I. J. Differential Geom. 33, 635–681MathSciNetMATHGoogle Scholar
  6. [6]
    Falcone, M., Ferretti, R. (2002): Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175, 559–575MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Grayson, M.A. (1989): A short note on the evolution of a surface via its mean curvature. Duke Math. J. 58, 555–558MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Kloeden, RE., Platen, E. (1992): Numerical solution of stochastic differential equations. Springer, BerlinMATHGoogle Scholar
  9. [9]
    Leoni, F. (2001): Convergence of an approximation scheme for curvature-dependent motions of sets. SIAM J. Numer. Anal. 39, 1115–1131MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Merriman, B., Bence, J., Osher, S. (1992): Diffusion generated motion by mean curvature. In: Taylor, J. (ed.): Computational crystal growers workshop. American Mathematical Society, Providence, RIGoogle Scholar
  11. [11]
    Merriman, B., Bence, J., Osher, S. (1994): Motion of multiple functions: a level set approach. J. Comput. Phys. 112, 334–363MathSciNetCrossRefGoogle Scholar
  12. [12]
    Osher, S., Fedkiw, R.P. (2001): Level-set methods: an overview and some recent results. J. Comput. Phys. 169, 463–502MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Osher, S., Sethian, J.A. (1988): Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Sethian, J.A. (1996): Level set methods. Cambridge University Press, CambridgeMATHGoogle Scholar
  15. [15]
    Soner, H.M., Touzi, N. (2001): A stochastic representation for mean curvature type flows. Ann. Probab., to appearGoogle Scholar
  16. [16]
    Souganidis, P.E. (1997): Front propagation: theory and applications. In: Bardi, M. et al. (eds.): Viscosity solutions and applications. (Lecture Notes in Mathematics, vol. 1660). Springer, Berlin, pp. 186–242CrossRefGoogle Scholar
  17. [17]
    Staniforth, A.N., Côté, J. (1991): Semi-Lagrangian integration schemes for atmospheric models — a review. Monthly Weather Rev. 119, 2206–2223CrossRefGoogle Scholar
  18. [18]
    Strain, J. (1999): Semi-Lagrangian methods for level set equations. J. Comput. Phys. 151, 498–533MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • M. Falcone
    • 1
  • R. Ferretti
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di MatematicaUniversitá di Roma TreRomaItaly

Personalised recommendations