A convergent monotone difference scheme for motion of level sets by mean curvature
- 202 Downloads
An explicit convergent finite difference scheme for motion of level sets by mean curvature is presented. The scheme is defined on a cartesian grid, using neighbors arranged approximately in a circle. The accuracy of the scheme, which depends on the radius of the circle, dx, and on the angular resolution, dθ, is formally O(dx2+dθ). The scheme is explicit and nonlinear: the update involves computing the median of the values at the neighboring grid points. Numerical results suggest that despite the low accuracy, acceptable results are achieved for small stencil sizes. A numerical example is presented which shows that the centered difference scheme is non-convergent.
KeywordsGrid Point Finite Difference Mathematical Method Difference Scheme Centered Difference
Unable to display preview. Download preview PDF.
- 4.Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27(1), 1–67 (1992)Google Scholar
- 6.Michael G. Crandall. Viscosity solutions: a primer. In: Viscosity solutions and applications (Montecatini Terme, 1995), Springer, Berlin, 1997, pp. 1–43Google Scholar
- 9.Evans, L.C.: Partial differential equations. American Mathematical Society, Providence, RI, 1998Google Scholar
- 12.Oberman, A.M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SINUM, To appear 2005Google Scholar
- 13.Osher, S., Fedkiw, R.: Level set methods and dynamic implicit surfaces. volume 153 of Appl. Math. Sci. Springer-Verlag, New York, 2003Google Scholar
- 14.Osher, S., Paragios, N.: (eds.), Geometric level set methods in imaging, vision, and graphics. Springer-Verlag, New York, 2003Google Scholar
- 16.Sethian, J.A.: Level set methods and fast marching methods. volume 3 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, second edition, 1999. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science.Google Scholar