Curvature Flow an Advanced Example

This lecture covers slides 12 .

Reading Material

Introduction to Mean Curvature Flow

This video will

  • Introduce the mean curvature flow problem and the partial differential equation (PDE) that describes this flow problem
  • Test yourself: Explain to the person sitting next to you intuitively what a mean curvature flow does, what do you think it will be useful for?
  • Covers: slides 12 (pages 3-5)

Mathematical Rewriting The Mean Curvature Flow Problem

This video will

  • Show how to rewrite the curvature term of the mean curvature flow PDE
  • Test yourself: Why does one do this rewrite? Is it not just a lot of fun math for nothing? The two curvature formulas are algebraic equivalent? Try and find good reasons for the rewrite.
  • Covers: slides 12 (pages 6-7)

Spatial Discretization of Mean Curvature Term

This video will

  • Walk through the spatial discretization steps of the  mean curvature flow curvature term
  • Covers: slides 12 (pages 8-13)

Handling Numerical "Side-effects" 

This video will

  • Show how to deal with numerical considerations such as noise, round-off errors and other things that can destroy your computations
  • Test yourself: Make some illustrations of contour shapes on a grid that will cause the numerical issues that are covered in the video. Try and account for the nature that causes the issues to pop up. 
  • Covers: slides 12 (pages 14-16)

Adding Temporal Discretization

The video will

  • Show how to do the temporal discretization of the  mean curvature flow temporal derivative
  • Test yourself: Try and explain to the person next to you what the intuition is behind the CFL condition. Exactly what is the main idea behind this condition? Try and come up with a flow example problem where a CFL condition would be necessary in order to get the numerical solution correct.
  • Covers: slides 12 (pages 17-19)