Variational discretization of axisymmetric curvature flows

TL;DR

This paper introduces and analyzes new axisymmetric numerical schemes for curvature flows, demonstrating their stability, efficiency, and suitability for complex geometries in a mostly one-dimensional setting.

Contribution

The authors develop and analyze novel axisymmetric variants of curvature flow schemes, providing stability, existence, and uniqueness results, which were previously lacking in the literature.

Findings

Schemes are stable and uniquely solvable.

Numerical results confirm high efficiency and accuracy.

Methods effectively handle complex geometries.

Abstract

We present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail. Although numerical methods for geometric flows have been used frequently in axisymmetric settings, numerical analysis results so far are rare. In this paper, we present stability, equidistribution, existence and uniqueness results for the introduced approximations. Numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one-dimensional problem has to be solved. The good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries.