Skip to main content
SpringerLink
Log in

An Efficient Boundary Integral Scheme for the MBO Threshold Dynamics Method via the NUFFT

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The MBO threshold dynamics method consists of two steps. The first step solves a pure initial value problem of the heat equation with the initial data being the indicator function of some bounded domain. In the second step, the new sharp interface is generated via thresholding either by some prescribed solution value or by volume preserving. We propose an efficient boundary integral scheme for simulating the threshold dynamics via the nonuniform fast Fourier transform (NUFFT). The first step is carried out by evaluating a boundary integral via the NUFFT, and the second step is performed applying a root-finding algorithm along the normal directions of a discrete set of points at the interface. Unlike most existing methods where volume discretization is needed for the whole computational domain, our scheme requires the discretization of physical space only in a small neighborhood of the interface and thus is meshfree. The algorithm is spectrally accurate in space for smooth interfaces and has \(O(N\log N)\) complexity, where N is the total number of discrete points near the interface when the time step \(\Delta t\) is not too small. The performance of the algorithm is illustrated via several numerical examples in both two and three dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Barles, G., Georgelin, C.: A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32(2), 484–500 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chambolle, A., Novaga, M.: Convergence of an algorithm for the anisotropic and crystalline mean curvature flow. SIAM J. Math. Anal. 37(6), 1978–1987 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numerica 14, 139–232 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14, 1368–1393 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. II. Appl. Comput. Harmon. Anal. 2, 85–100 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic Press, Cambridge (1972)

    MATH  Google Scholar 

  7. Esedoglu, S., Otto, F.: Threshold dynamics for networks with arbitrary surface tensions. Comm. Pure Appl. Math. 68, 808–864 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Esedoglu, S., Ruuth, S., Tsai, R.: Threshold dynamics for high order geometric motions. Interf. Free Bound. 10(3), 263–282 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Esodoglu, S., Smereka, P.: A variational formulation for a level set representation of multiphase flow and area preserving curvature flow. Comm. Math. Sci. 6(1), 125–148 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Evans, L.C.: Convergence of an algorithm for mean curvature motion. Indiana Math. J. 42(2), 533–557 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  12. Gao, M., Wang, X.: A gradient stable scheme for a phase field model for the moving contact line problem. J. Comput. Phys. 231, 1372–386 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Greengard, L., Lee, J.Y.: Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46(3), 443–454 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9(1), 83–97 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Greengard, L., Strain, J.: The fast Gauss transform. SIAM J. Sci. Statist. Comput. 12(1), 79–94 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Greengard, L., Sun, X.: A new version of the fast Gauss transform. Proc. Int. Cong. Math. III, 575–584 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Ilmanen, T.: Lectures on Mean Curvature Flow and Related Equations, Lecture Notes. ICTP, Trieste (1995). http://www.math.ethz.ch/~ilmanen/papers/pub.html

  18. Ishii, K.: Optimal rate of convergence of the Bence–Merriman–Osher algorithm for motion by mean curvature. SIAM J. Math. Anal. 37(3), 841–866 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kublik, C., Esedoglu, S., Fessler, J.A.: Algorithms for area preserving flows. SIAM J. Sci. Comput. 33(5), 2382–2401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lee, J.Y., Greengard, L., Gimbutas, Z.: NUFFT Version 1.3.2 Software Release. http://www.cims.nyu.edu/cmcl/nufft/nufft.html (2009)

  21. Leung, S., Zhao, H.: A grid based particle method for moving interface problems. J. Comput. Phys. 228(8), 2993–3024 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, J.R., Greengard, L.: On the numerical solution of the heat equation. I. Fast solvers in free space. J. Comput. Phys. 226(2), 1891–1901 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mascarenhas, P.: Diffusion Generated Motion by Mean Curvature. Department of Mathematics. University of California, Los Angeles, Los Angeles (1992)

    Google Scholar 

  24. Merriman, B., Bence, J.K., Osher, S.: Diffusion generated motion by mean curvature. UCLA CAM Report 92-18 (1992)

  25. Müller, D.E.: A method for solving algebraic equations using an automatic computer. Math. Tables Aids Comput. 10, 208–215 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mullins, W.W.: Two-dimensional motion of idealized grain boundaries. J. Appl. Phys. 27(8), 900–904 (1956)

    Article  MathSciNet  Google Scholar 

  27. Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Randol, B.: On the Fourier transform of the indicator function of a planar set. Trans. Amer. Math. Soc. 139, 271–278 (1969)

    MathSciNet  MATH  Google Scholar 

  29. Ruuth, S.: An algorithm for generating motion by mean curvature. ICAOS’96 pp. 82–91 (1996)

  30. Ruuth, S.J.: Efficient algorithms for diffusion-generated motion by mean curvature. Ph.D. thesis, University of British Columbia, Vancouver, Canada (1996)

  31. Ruuth, S.J.: A diffusion-generated approach to multiphase motion. J. Comput. Phys. 145(1), 166–192 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ruuth, S.J.: Efficient algorithms for diffusion-generated motion by mean curvature. J. Comput. Phys. 144(2), 603–625 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ruuth, S.J., Wetton, B.T.: A simple scheme for volume-preserving motion by mean curvature. J. Sci. Comput. 19(1–3), 373–384 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 23(2), 165–224 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  35. Svadlenka, K., Ginder, E., Omata, S.: A variational method for multiphase volume-preserving interface motions. J. Comput. Appl. Math. 257, 157–179 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  37. Womble, D.E.: A front-tracking method for multiphase free boundary problems. SIAM J. Numer. Anal. 26(2), 380–396 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xu, X., Wang, D., Wang, X.P.: An efficient threshold dynamics method for wetting on rough surfaces. J. Comput. Phys. 330, 510–528 (2017)

    Article  MathSciNet  Google Scholar 

  39. Yue, P., Feng, J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhao, H., Chan, T., Merriman, B., Osher, S.J.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

S. Jiang was supported by the National Science Foundation under Grant DMS-1418918. X. P. Wang was supported in part by the Hong Kong RGC-GRF Grants 605513 and 16302715, RGC-CRF Grant C6004-14G, and NSFC-RGC joint research Grant N-HKUST620/15.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    Shidong Jiang

  2. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    Dong Wang & Xiao-Ping Wang

Authors
  1. Shidong Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-Ping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shidong Jiang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, S., Wang, D. & Wang, XP. An Efficient Boundary Integral Scheme for the MBO Threshold Dynamics Method via the NUFFT. J Sci Comput 74, 474–490 (2018). https://doi.org/10.1007/s10915-017-0448-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0448-1

Keywords

Mathematics Subject Classification

Search

Navigation