2019 Fiscal Year Annual Research Report
Hyperbolic threshold dynamics: applications and analysis
| 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 / curvature flow / approximation methods |
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| Outline of Annual Research Achievements |
We developed a generalized threshold dynamical (TD) algorithm for approximating interfacial motions under the damped hyperbolic mean curvature flow. Our approximation method's numerical counterpart was used to perform computational tests which, in turn, confirmed the presence and influence of the damping term. Furthermore, we found that our TD algorithm is able to approximate motion by the standard parabolic mean curvature flow. In this sense, we succeeded to generalize the well-known MBO algorithm. These facts were also confirmed via numerical experiments.
Additionally, we established a minimizing movement for use within our hyperbolic threshold dynamics (TD). Our method is based on the minimization of time-discretized functionals of Crank-Nicolson type, and we showed that the minimizing movement is energy preserving. We also constructed the corresponding numerical method, and our computational results confirmed our analytical findings. In particular, since energy preservation is an essential property when performing simulations of the hyperbolic mean curvature flow, we showed that our method indeed imparts this property on the TD. Moreover, through the establishment of uniform energy estimates, we showed the existence, regularity, and convergence of minimizers.
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