| Speaker: Eszter Fehér Affiliation: Budapest University of Technology and Economics Time: 2 pm Thursday, 16 May, 2019 Location: 303-257 |
| Curvature-driven flows are a broad class of geometric partial differential equations. In the planar case they describe the evolution of a curve the points of which move in the normal direction with speed v defined as some function of the curvature k. The simplest such flow is given as v=k and is called the curve shortening flow, first proposed by Firey as a model the abrasion of colliding stones. Firey’s model was broadly generalized both mathematically (Andrews) and geophysically (Bloore), resulting in what we call the Andrews-Bloore flow. Although the equation can be treated with traditional numerical techniques (such as the level-set method), these algorithms fail to capture the fine geometrical properties of the flow. In particular, the time evolution of geophysical shape descriptors is of key interest to geophysicists and one of the most important shape descriptor is the number N of critical points if the curve is interpreted as a scalar distance from a fixed point or from the center of mass. While the evolution N(t) is of central importance in geomorphology, it is particularly sensitive to numerical errors as it relies on derivatives, and so far no reliable algorithm has been available to compute it. In my talk I will describe a front-tracking algorithm using a polynomial representation of the curve which we developed with the main goal to investigate N(t) in the Andrews-Bloore flows. I will demonstrate the stability and robustness of the algorithm partly on selected numerical examples, partly by verifying known theoretical results on N(t) which have not been checked computationally before. We also tested some curious mathematical conjectures. |