Multisymplectic Geometry and Geometric Numerical Integrator for Variational Problems
| Project/Area Number |
20K14365
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Keio University |
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Principal Investigator |
彭 林玉 慶應義塾大学, 理工学部(矢上), 准教授 (90725780)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2023: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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| Keywords | Moving frame / Geometric integration / Symmetry-preserving / Burgers' equation / Kepler problem / Symplectic structure / Euclidean group / Formal Lagrangian / Variational integrator / KdV equation / Symmetry reduction / Group-invariant solution / Similarity solution / Soliton / Variational principle / Noether's theorem / Symmetry / Conservation law / DPD / Hamiltonian system / 修正形式ラグランジアン / ハミルトン偏微分方程式 / Geometric integrator / Variational bicomplex / Variational problems / PDEs |
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| Outline of Research at the Start |
Geometric integrator is among one of the most efficient numerical methods for differential equations. In this project, we establish a unified and systematical analogue for understanding both continuous and discrete multisymplectic structures of arbitrary order variational differential equations.
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| Outline of Annual Research Achievements |
Firstly, the modified formal Lagrangian structure was applied to practical problems. In particular, we extensively studied various numerical solutions of the viscous Burgers' equation, including shock waves. These show improvement of error in the aspect of conservation law-preserving property. Further applications will be conducted, for instance, novel numerical methods for Euler and Navier-Stokes equations.
At the same time, we further investigated the application of discrete moving frames for constructing symmetry-preserving (and hence conservation law-preserving from Noether's theorem) numerical methods for mechanical systems. Following our previous results on Euler's elastica that preserve SE(3) symmetries, we are currently finalizing a similar work towards SE(3)-preserving numerical methods for the Kepler problem. This approach allows us to preserve the total energy and symplectic structures simultaneously, although the step size may vary.
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Report
(4 results)
Research Products
(55 results)
