Numerical analysis for support splitting and merging phenomena appearing in the interfacial dynamics

• Project/Area Number: 16K05271
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyoto University
• Principal Investigator: Tomoeda Kenji 京都大学, 情報学研究科, 研究員 (60033916)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 数値解析 / 差分法 / 界面ダイナミクス / サポート分離併合 / 零点集合の挙動 / 自由境界

Outline of Final Research Achievements

In this study we have the following numerical results, which is concerned with the realization of the behaviour of the non-stationary seepage governed by non-linear diffusion and evaporation.
(1) Imposing the constant value on the boundary condition, we observed that the numerical support which is a simple connected domain changes into a non-connected domain as the time increases. This means the appearance of the zeros of the solution of the model equation. (2) Imposing the sufficiently long period on it, we obtained the phenomena of the repeated appearance and disappearance of the zeros. (3) On the other hand, imposing the sufficiently short period on it, we didn't observe the appearance of the zeros.
We proved the phenomena stated above from the mathematical point of views.

Academic Significance and Societal Importance of the Research Achievements

「蒸発過程での浸透領域の変化」とは 海に囲まれた島の内陸部では分布する流体の蒸発と拡散及び島の周囲からの海水の浸入・流出によって引き起こされる現象である。現実の現象は極めて複雑であるので その第一歩として最も簡単な数理モデルを構築し その定性的な性質を調べるという基礎研究に着手した。その結果 周期的な潮位による浸入・流出によって 浸透領域が島全体に分布したり又は 一部に非浸透領域が現れることが数値・数学解析の観点から得られた。これは地域を浸入水から守る意味での知見として意味があるのではないかと思われる。

Research Products

• [Journal Article] Behaviour of the support of the solution appearing in some nonlinear diffusion equation with absorption (2017)
  • Author(s): Kenji Tomoeda
  • Journal Title: Proceedings of EQUADIFF 2017 Conference, Published by Slovak University of Technology, SPEKTRUM STU Publishing Volume: EQUADIFF 2017 Pages: 359-368
  • Peer Reviewed
• [Presentation] Numerical approach to the repeated appearance and disappearance phenomena of non-infiltrated area with fluid through porous media (2019)
  • Author(s): Kenji Tomoeda
  • Organizer: 9th International Congress on Industrial and Applied Mathematics - ICIAM 2019, Valencia, Spain
  • Int'l Joint Research
• [Presentation] Appearance and disappearance of the region occupied by the flow through a boundary (2019)
  • Author(s): Kenji Tomoeda
  • Organizer: ANZIAM (Australia and New Zealand Industrial and Applied Mathematics) 2019, Nelson, New Zealand
  • Int'l Joint Research
• [Presentation] Appearance and disappearance of the region infiltrated with fluid in an absorbing medium (2018)
  • Author(s): Kenji Tomoeda
  • Organizer: International Conference Czech-Japanese Seminar in Applied Mathematics 2018, Noto-cho, Japan
  • Int'l Joint Research
• [Presentation] Van der Pol方程式とその真空管回路 ～ そのモデル化の背景 ～ (2018)
  • Author(s): 友枝 謙二
  • Organizer: 北陸応用数理研究会 2018
• [Presentation] 吸収項を伴う多孔性媒体流における浸透領域の分離・併合について (2017)
  • Author(s): 友枝謙二
  • Organizer: 研究集会「数値解析の理論と実践」
  • Place of Presentation: 石川県政記念 しいのき迎賓館
  • Year and Date: 2017-03-21
• [Presentation] Support splitting and merging phenomena in the solution of porous media equations (2016)
  • Author(s): Kenji Tomoeda
  • Organizer: International Seminar on Applied Mathematics for Real-world Problems II
  • Place of Presentation: Hokkaido University
  • Year and Date: 2016-10-28