Journal of Scientific Computing

, Volume 35, Issue 2–3, pp 219–240 | Cite as

Level Set Equations on Surfaces via the Closest Point Method

  • Colin B. Macdonald
  • Steven J. Ruuth


Level set methods have been used in a great number of applications in ℝ2 and ℝ3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in ℝ3 or higher dimensions. In this paper we consider the treatment of level set equations on surfaces via a recent technique for solving partial differential equations (PDEs) on surfaces, the Closest Point Method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943–1961, [2008]). Our main modification is to introduce a Weighted Essentially Non-Oscillatory (WENO) interpolation step into the Closest Point Method. This, in combination with standard WENO for Hamilton–Jacobi equations, gives high-order results (up to fifth-order) on a variety of smooth test problems including passive transport, normal flow and redistancing. The algorithms we propose are straightforward modifications of standard codes, are carried out in the embedding space in a well-defined band around the surface and retain the robustness of the level set method with respect to the self-intersection of interfaces. Numerous examples are provided to illustrate the flexibility of the method with respect to geometry.


Closest Point Method Level set methods Partial differential equations Implicit surfaces WENO schemes WENO interpolation 


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  1. 1.
    Bertalmío, M., Cheng, L.-T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759–780 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cheng, L.-T., Tsai, R.: Redistancing by flow of the time dependent Eikonal equation (2008). Under review Google Scholar
  3. 3.
    Cheng, L.-T., Burchard, P., Merriman, B., Osher, S.: Motion of curves constrained on surfaces using a level-set approach. J. Comput. Phys. 175(2), 604–644 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Crandall, M.G., Lions, P.-L.: Two approximations of solutions of Hamilton–Jacobi equations. Math. Comput. 43(167), 1–19 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152(2), 457–492 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Greer, J.B.: An improvement of a recent Eulerian method for solving PDEs on general geometries. J. Sci. Comput. 29(3), 321–352 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Laney, C.B.: Computational Gasdynamics. Cambridge University Press, Cambridge (1998) zbMATHGoogle Scholar
  10. 10.
    Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Merriman, B., Ruuth, S.J.: Diffusion generated motion of curves on surfaces. J. Comput. Phys. 225(2), 2267–2282 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Merriman, B., Ruuth, S.J.: Embedding methods for the numerical solution of PDEs on manifolds. In preparation Google Scholar
  13. 13.
    Mitchell, I.: A toolbox of level set methods. Technical Report TR-2004-09, University of British Columbia Department of Computer Science, July 2004.
  14. 14.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2003) zbMATHGoogle Scholar
  15. 15.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Russo, G., Smereka, P.: A remark on computing distance functions. J. Comput. Phys. 163(1), 51–67 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ruuth, S.J., Merriman, B.: A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys. 227(3), 1943–1961 (2008) zbMATHCrossRefGoogle Scholar
  19. 19.
    Saboret, L., Attene, M., Alliez, P.: “Laurent’s Hand”, the AIM@SHAPE shape repository (2007).
  20. 20.
    Sebastian, K., Shu, C.-W.: Multidomain WENO finite difference method with interpolation at subdomain interfaces. J. Sci. Comput. 19(1–3), 405–438 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge Monographs on Applied and Computational Mathematics, vol. 3, 2nd edn. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  22. 22.
    Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Technical Report NASA CR-97-206253 ICASE Report No. 97-65, Institute for Computer Applications in Science and Engineering, November 1997 Google Scholar
  23. 23.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wikipedia contributors: Klein bottle. Wikipedia, the free encyclopedia, (2007). Accessed 29 May 2007

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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