Hyperbolic threshold dynamics: applications and analysis

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K14229
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Meiji University
• Principal Investigator: Ginder Elliott 明治大学, 総合数理学部, 専任准教授 (30648217)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: Threshold dynamics / interfacial motion / approximation methods / curvature flow

Outline of Final Research Achievements

The main result of this research is the discovery of a generalized MBO algorithm (GMBO) using hyperbolic threshold dynamics. The GMBO was found to be an approximation method for the damped hyperbolic mean curvature flow. The result enables one to approximate oscillatory interfacial motions, as well as mean curvature flow. In particular, by changing the base PDE used in the MBO to the wave equation, we were showed that the initial velocity field can be used to control the propagation of interfaces. Interestingly, this clarified that damped interfacial motions are not obtained through additional damping terms in the PDE, but by encoding normal velocity fields within level set functions at each time step. We also established working numerical methods for performing computational analyses and simulations. Here we developed an energy preserving minimizing movement for treating the hyperbolic mean curvature flow, and we confirmed the method’s properties through computational investigations.

Free Research Field

応用数学

Academic Significance and Societal Importance of the Research Achievements

今までの Threshold Dynamics 研究では，本研究のHMBO以外，MBO法に厳守されていた．MBOは，Level Set 法を生み出したものとして知られていたが，本研究の初期段階で作成したHMBOは，慣性の影響を含む振動する界面においてTDの適応範囲を広げることができた．GMBOはMBOとHMBOを同時に取り扱えるため，応用数学と産業に関する課題が１つのTD法で表現することが可能となった．また，今までのMinimizing Movementsの研究において，エネルギー保存するのものが不在だったため，本研究のCrank-Nicholson MMは異分野にも応用があると期待している．