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Convergence of a semi-discrete scheme for the curve shortening flow. (English) Zbl 0811.65112

Summary: Convergence for a spatial discretization of the curvature flow for curves in possibly higher codimension is proved in \(L^ \infty((0,T), L^ 2(\mathbb{R}/2\pi))\cap L^ 2((0,T), H^ 1(\mathbb{R}/2\pi))\). Asymptotic convergence in these norms is achieved for the position vector and its time derivative which is proportional to curvature. The underlying algorithm rests on a formulation of mean curvature flow which uses the Laplace-Beltrami operator and leads to tridiagonal linear systems which can easily be solved.

MSC:

65Z05 Applications to the sciences
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
37C80 Symmetries, equivariant dynamical systems (MSC2010)
35K15 Initial value problems for second-order parabolic equations
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