zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Second-order accurate computation of curvatures in a level set framework using novel high-order reinitialization schemes. (English) Zbl 1203.65043
Summary: We present a high-order accurate scheme for the reinitialization equation of {\it M. Sussman, P. Smereka} and {\it S. Osher} [J. Comput. Phys. 114, No. 1, 146--159 (1994; Zbl 0808.76077)] that guarantees accurate computation of the interface’s curvatures in the context of level set methods. This scheme is an extension of the work of {\it G. Russo} and {\it P. Smereka} [J. Comput. Phys. 163, No. 1, 51--67 (2000; Zbl 0964.65089)]. We present numerical results in two and three spatial dimensions to demonstrate fourth-order accuracy for the reinitialized level set function, third-order accuracy for the normals and second-order accuracy for the interface’s mean curvature in the $L ^{1}$- and $L ^{\infty }$-norms. We also exploit the work of {\it Ch. Min} and {\it F. Gibou} [J. Comput. Phys. 225, No. 1, 300--321 (2007; Zbl 1122.65077)] to show second-order accurate scheme for the computation of the mean curvature on non-graded adaptive grids.

65D18Computer graphics, image analysis, and computational geometry
53C44Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Full Text: DOI
[1] Gibou, F., Fedkiw, R.: A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. Comput. Phys. 202, 577--601 (2005) · Zbl 1061.65079 · doi:10.1016/j.jcp.2004.07.018
[2] Gibou, F., Fedkiw, R., Caflisch, R., Osher, S.: A level set approach for the numerical simulation of dendritic growth. J. Sci. Comput. 19, 183--199 (2003) · Zbl 1081.74560 · doi:10.1023/A:1025399807998
[3] Gibou, F., Levy, D., Cardenas, C., Liu, P., Boyer, A.: Partial differential equation based segmentation for radiotherapy treatment planning. Math. Biosci. Eng. 2, 209--226 (2005) · Zbl 1070.92024
[4] Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126--2143 (2000) · Zbl 0957.35014 · doi:10.1137/S106482759732455X
[5] Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 126, 202--212 (1996) · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130
[6] Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. ACM Trans. Graph. (SIGGRAPH Proc.) 457--462 (2004)
[7] Losasso, F., Fedkiw, R., Osher, S.: Spatially adaptive techniques for level set methods and incompressible flow. Comput. Fluids 35, 995--1010 (2006) · Zbl 1177.76295 · doi:10.1016/j.compfluid.2005.01.006
[8] Min, C., Gibou, F.: A second order accurate level set method on non-graded adaptive Cartesian grids. J. Comput. Phys. 225, 300--321 (2007) · Zbl 1122.65077 · doi:10.1016/j.jcp.2006.11.034
[9] Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002) · Zbl 1026.76001
[10] Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, New York (2003) · Zbl 1027.68137
[11] Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12--49 (1988) · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[12] Russo, G., Smereka, P.: A remark on computing distance functions. J. Comput. Phys. 163, 51--67 (2000) · Zbl 0964.65089 · doi:10.1006/jcph.2000.6553
[13] Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999) · Zbl 0929.65066
[14] Shin, S.: Computation of the curvature field in numerical simulation of multiphase flow. J. Comput. Phys. 222, 872--878 (2007) · Zbl 1158.76411 · doi:10.1016/j.jcp.2006.08.009
[15] Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439--471 (1988) · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5
[16] Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32--78 (1989) · Zbl 0674.65061 · doi:10.1016/0021-9991(89)90222-2
[17] Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146--159 (1994) · Zbl 0808.76077 · doi:10.1006/jcph.1994.1155