2017 Fiscal Year Research-status Report
双曲型 Threshold Dynamics：応用と数理解析
| Project/Area Number |
17K14229
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| Research Institution | Meiji University |
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Principal Investigator |
Ginder Elliott 明治大学, 総合数理学部, 専任准教授 (30648217)
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | threshold dynamics / interfacial motion / hyperbolic pde / curvature flow / MBO |
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| Outline of Annual Research Achievements |
Using the hyperbolic MBO (HMBO) as a foundation, we have discovered a generalized threshold dynamical algorithm (TD). The generalization enables one to included damping terms into the approximation of interfacial dynamics. Similar to the original HMBO, the new TD algorithm is based on evolving interfaces using wave propagation. Here we were able to show, by prescribing a special initial velocity to the governing partial differential equation, that the interface can inherit a prescribed damping term. As a corollary, by taking a zero initial condition for the partial differential equation, we found that our initial velocity can realize the original MBO algorithm. In particular, our method is able to approximate motion by mean curvature flow and includes the MBO as a special case for certain parameter choices.
Corresponding to these results, we also constructed numerical algorithms and performed error analyses in two and three dimensions. We found, in both cases, that our analytic results are in agreement with our computational observations.
In addition, we began researching candidate TD algorithms for surface constrained interfacial motions. A formal approximation method for surface constrained curvature flow was designed, and we investigated its numerical properties. Here, we succeeded to define the interface in a way that allows one to computationally track its evolution in time.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
Our advances have enabled us to include damping terms in the hyperbolic mean curvature flow. This represents is a fundamental control and greatly enhances the range of applications of our TD algorithms to the approximation of other interfacial motions. Moreover, our numerical analyses support our theoretic results. By showing that the initial velocity of the wave equation can control interfacial damping, this research has also shed new light on the role that signed distance functions play in approximating hyperbolic interfacial motions utilizing level set frameworks. Most importantly, our analysis has also unearthed interesting questions and promising directions for further research.
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| Strategy for Future Research Activity |
The mathematical analysis of the hyperbolic MBO and its generalization will continue. Here, we are immediately interested in providing an elementary understanding of how the generalized TD algorithm can incorporate the MBO within its framework. We also anticipate that this research can further generalize to treat the setting of surface constrained interfacial motions, for which we aim to establish the corresponding mathematical frameworks. In addition to the above, we would like to illustrate the use of our TD algorithms within coupled models (e.g., where interfaces move within external velocity fields) and to continue performing numerical investigations of our TD algorithms.
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| Causes of Carryover |
Travel fees decreased significantly upon transferring to new university. The remaining funds will be used in establishing our laboratory's web server.
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Research Products
(1 results)
