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

Abstract

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.

Evolution of plane curves with a curvature adjusted tangential velocity

“…In order to derive a discrete numerical scheme, we follow the flowing finite volume method adopted for curve evolutionary problems as it was proposed by Mikula et al in [17,19]. Let us introduce the "dual" volume S…”

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

“…If α eff > 1, then (α eff − 1)κ induces instability, and δκ ss plays a stabilization role of the unstable front. An alternative stabilization method is to use the Willmore flow [5]. Note that the tangential velocity W has no effect on the shape of evolving front, which is determined by the value of the normal velocity V only.…”

“…Step 3 To define the curvatures on P i and at x i , we use (5) rather than the Frenet formulae, i.e., we recall that the curvature can be defined by the first variation of the total length L from (5). From (7), the total length L, andṙ i =…”

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

“…Often, operator splitting is employed, thus enabling the use of H 1 conforming spaces. But also more direct approaches exist, for instance, using finite volume techniques as in [33], employing methods from isogeometric analysis ( [6]), or using C 1 conforming finite elements as in [15,14]. Alternatively, methods may also be based on interface capturing approaches.…”