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A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations
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Year:    2010

Author:    Karol Mikula, Daniel Ševčovič, Martin Balažovjech

Communications in Computational Physics, Vol. 7 (2010), Iss. 1 : pp. 195–211

Abstract

A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.2009.08.169

Communications in Computational Physics, Vol. 7 (2010), Iss. 1 : pp. 195–211

Published online:    2010-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    17

Keywords:   

Author Details

Karol Mikula

Daniel Ševčovič

Martin Balažovjech

  1. Inflow-implicit/outflow-explicit finite volume methods for solving advection equations

    Mikula, Karol | Ohlberger, Mario | Urbán, Jozef

    Applied Numerical Mathematics, Vol. 85 (2014), Iss. P.16

    https://doi.org/10.1016/j.apnum.2014.06.002 [Citations: 16]

  2. NaturaSat—A Software Tool for Identification, Monitoring and Evaluation of Habitats by Remote Sensing Techniques

    Mikula, Karol | Šibíková, Mária | Ambroz, Martin | Kollár, Michal | Ožvat, Aneta A. | Urbán, Jozef | Jarolímek, Ivan | Šibík, Jozef

    Remote Sensing, Vol. 13 (2021), Iss. 17 P.3381

    https://doi.org/10.3390/rs13173381 [Citations: 11]

  3. New fast and stable Lagrangean method for image segmentation

    Mikula, Karol | Urban, Jozef

    2012 5th International Congress on Image and Signal Processing, (2012), P.688

    https://doi.org/10.1109/CISP.2012.6469852 [Citations: 6]

  4. A simple and fast numerical method for solving flame/smoldering evolution equations

    Goto, Maika | Kuwana, Kazunori | Yazaki, Shigetoshi

    JSIAM Letters, Vol. 10 (2018), Iss. 0 P.49

    https://doi.org/10.14495/jsiaml.10.49 [Citations: 6]

  5. Evolution of plane curves with a curvature adjusted tangential velocity

    Ševčovič, Daniel | Yazaki, Shigetoshi

    Japan Journal of Industrial and Applied Mathematics, Vol. 28 (2011), Iss. 3 P.413

    https://doi.org/10.1007/s13160-011-0046-9 [Citations: 21]

  6. A Higher Order Scheme for a Tangentially Stabilized Plane Curve Shortening Flow with a Driving Force

    Balažovjech, Martin | Mikula, Karol

    SIAM Journal on Scientific Computing, Vol. 33 (2011), Iss. 5 P.2277

    https://doi.org/10.1137/100795309 [Citations: 17]
  7. Scale Space and Variational Methods in Computer Vision

    3D Curve Evolution Algorithm with Tangential Redistribution for a Fully Automatic Finding of an Ideal Camera Path in Virtual Colonoscopy

    Mikula, Karol | Urbán, Jozef

    2012

    https://doi.org/10.1007/978-3-642-24785-9_54 [Citations: 4]

  8. A new tangentially stabilized 3D curve evolution algorithm and its application in virtual colonoscopy

    Mikula, Karol | Urbán, Jozef

    Advances in Computational Mathematics, Vol. 40 (2014), Iss. 4 P.819

    https://doi.org/10.1007/s10444-013-9328-x [Citations: 4]

  9. Curve shortening flow coupled to lateral diffusion

    Pozzi, Paola | Stinner, Björn

    Numerische Mathematik, Vol. 135 (2017), Iss. 4 P.1171

    https://doi.org/10.1007/s00211-016-0828-8 [Citations: 14]

  10. Macrophages Trajectories Smoothing by Evolving Curves

    Lupi, Giulia | Mikula, Karol | Park, Seol Ah

    Tatra Mountains Mathematical Publications, Vol. 0 (2023), Iss. 0

    https://doi.org/10.2478/tmmp-2023-0031 [Citations: 0]

  11. Elastic flow interacting with a lateral diffusion process: the one-dimensional graph case

    Pozzi, Paola | Stinner, Björn

    IMA Journal of Numerical Analysis, Vol. (2018), Iss.

    https://doi.org/10.1093/imanum/dry004 [Citations: 0]