2022 Fiscal Year Final Research Report
HIgh-precision numerical analysis of fluid phenomena by the method of fundamental solutions
| Project/Area Number |
18K13455
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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 | Okayama University of Science (2020-2022)
Kyoto University (2018-2019) |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 基本解近似解法 / Hele-Shaw問題 / 極小曲面 / Plateau問題 |
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| Outline of Final Research Achievements |
The goal of this research was to apply the method of fundamental solutions, known as mesh-free numerical method for partial differential equations, to fluid phenomena. As a result, for the first time in the world, we proposed a spatial discretization for the Hele-Shaw problem that preserves its geometric variational structure in an asymptotic sense, and confirmed its usefulness through mathematical analysis and extensive numerical experiments. We also designed a fast and accurate algorithm for determining the number of minimal surfaces (in particular, the Plateau problem), proved its convergence to minimal surfaces, and demonstrated its usefulness in various concrete examples.
Translated with www.DeepL.com/Translator (free version)
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| Free Research Field |
偏微分方程式の数値解析
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| Academic Significance and Societal Importance of the Research Achievements |
基本解近似解法は，そのスキームの簡便さ故に主に工学分野で多く応用されてきたが，流体現象を記述する問題に対する応用にはあまり成功していなかった．本研究では，Hele-Shaw問題に対する応用により移動境界問題に対する基本解近似解法の有用性を実証することに成功した．また，高精度であることを活かした極小曲面の数値計算アルゴリズムを提唱しその解析に成功したことで，極小曲面の滑らかな近似を，理論保証込みで初めて可能とした．
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