Commun. Comput. Phys., 7 (2010), pp. 195-211.


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

Karol Mikula 1*, Daniel Sevcovic 2, Martin Balazovjech 1

1 Department of Mathematics, Slovak University of Technology, Radlinskeho 11, 813 68 Bratislava, Slovak Republic.
2 Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava, Slovak Republic.

Received 25 October 2008; Accepted (in revised version) 29 May 2009
Available online 29 July 2009
doi:10.4208/cicp.2009.08.169

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.

AMS subject classifications: 35K65, 65N40, 53C80, 35K55, 53C44, 65M60

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Key words: Geometric partial differential equations, evolving plane curves, mean curvature flow, anisotropy, Willmore flow, surface diffusion, finite volume method, semi-implicit scheme, tangential redistribution.

*Corresponding author.
Email: mikula@math.sk (K. Mikula), sevcovic@fmph.uniba.sk (D. Sevcovic), balazoviech@math.sk (M. Balazovjech)
 

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