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), 講師 (90725780)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Project Status |
Granted (Fiscal Year 2022)
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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 | 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 / Symplectic structure / 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 |
We proposed the modified formal Lagrangian structure for arbitrary differential equations and applied it to the derivation of conservation laws using Noether’s theorem. This is also used to constructing (formal) variational integrator for nonvariational equations. We also analyzed novel wave structures of a variable-coefficient KdV system by Hirota’s bilinear method and symmetry analysis; a variety of solitons were obtained as well as novel third-order Painleve equations.
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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
As planned, we have developed Noether’s theorems for discrete equations and have proposed structure-preserving numerical methods for non-variational differential equations based on the modified formal Lagrangian formulation. Several papers were published in leading academic journals together with a couple of invited international and domestic conference/workshop presentations.
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| Strategy for Future Research Activity |
The research will be continued following the original proposal. We have been studying the emerging of geometric integrator with machine learning as well.
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Report
(3 results)
Research Products
(40 results)
