A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations (Communications in Computational Physics, 2010)
A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto 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. Theevolution is numerically stabilized by an asymptotically uniform tangential redistributionof grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numericallyapproximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.