https://neckpinch.com/authors/robert-nurnberg/New arXiv papers by Robert Nürnberg, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2204.12878/arxiv:2204.12878Wed, 27 Apr 2022 00:00:00 GMTKlaus Deckelnick, Robert Nürnberg — Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave phenomena in solid-liquid interfaces. We introduce a semidiscrete finite difference method for the approximation of hyperbolic curvature flow and prove error bounds for natural discrete norms. We also present numerical simulations, including the onset of singularities starting from smooth strictly convex initial data.Curve shorteningOther curvature flowsSingularitiesNumerics & visualization https://neckpinch.com/papers/2110.04605/arxiv:2110.04605Sat, 09 Oct 2021 00:00:00 GMTKlaus Deckelnick, Robert Nürnberg — We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to e.g.\ geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal H¹–error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.Curve shorteningOther curvature flowsAnisotropic & nonlocal flowsStability & uniquenessHarmonic maps & DeTurck trickNumerics & visualization