Numerical analysis of the chaotic bahavior of the free boundary

自由边界混沌行为的数值分析

• 批准号: 09440080
• 负责人: IMAI Hitoshi
• 金额: $ 2.11万
• 依托单位: The University of Tokushima
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440080/
• 关键词: free boundary; chaos; PVM; pattern; spectral method; bifurcation; phase separation; reaction-difusion; 高精度; 任意精度; 無限精度

In the term of the project (three years) many results are obtained. Important results are shown as follows.1. Development of numerical methods for analysis of chaotic phenomena.The group of Tokushima Univ. developed a method for analysis of chaotic phenomena in free boundary problems.It is based on numerical methods, so it is general. Moreover, it has a surprising feature that attractors in the infinite-dimensional solution space can be approximated arbitrarily. This means the method connects theoretical analysis and numerical analysis, therefore it enhances development of theoretical and numerical researches.2. A proposal of a free boundary problem with attractors.The group of Tokushima Univ. proposed a one-dimensional free boundary problem with some parameters. It has exact solutions for special values of the parameters, so it is covenient for the check of the method and programs. Various attractors are found numerically.3. Analysis of the pattern.Nishiura solved self-replicating dynamics of the dissipative system. He also revealed the minimizer of the system which describes the pattern formation of the diblock copolymer has the fine structure with the meso-scale. Sakaguchi proposed a model to formation of colony patterns by a bacterial cell population. From numerical simulation various patterns are observed in spite of the simplicity of the model.4. Development of fast numerical computation and its application.The group of Tokushima Univ., Ikeda and Okamoto developed methods for fast numerical computation in the environment of parallel computing offered by PVM. These methods enabled large-scaled numerical simulation of free boundary problems or fluid mechanics. Some new phenomena were found.

在该项目的期限（三年）内，取得了许多成果。主要结果如下：1.混沌现象分析的数值方法的发展德岛大学的研究小组发展了一种分析自由边界问题中混沌现象的方法，它是基于数值方法的，因此具有普遍性。此外，它有一个令人惊讶的特点，吸引子在无限维的解决方案空间可以任意近似。这意味着该方法将理论分析和数值分析结合起来，从而促进了理论和数值研究的发展.一个带吸引子的自由边界问题的提出。德岛大学的研究小组提出了一个带参数的一维自由边界问题。它对参数的特殊值有精确解，便于方法和程序的检验。各种吸引子被发现数字。3.模式分析。西浦解决了耗散系统的自我复制动力学。他还揭示了描述两嵌段共聚物图案形成的体系的极小化物具有介观尺度的精细结构。坂口提出了一个细菌细胞群形成菌落模式的模型。从数值模拟中观察到的各种模式，尽管简单的模型.快速数值计算的发展及其应用。德岛大学，Ikeda和Okamoto在PVM提供的并行计算环境中开发了快速数值计算的方法。这些方法使自由边界问题或流体力学的大规模数值模拟成为可能。发现了一些新的现象。

项目成果

今井仁司, 竹内敏己, 坂口秀雄, 篠原能材, Tarmizi: "第微分方程式の無限精度数値シミュレーションについて" 第47回応用力学連合講演会講演予稿集. 435-436 (1998)

Hitoshi Imai、Toshiki Takeuchi、Hideo Sakaguchi、Noki Shinohara、Tarmizi：“关于第一微分方程的无限精度数值模拟”第 47 届应用力学联盟会议记录 435-436 (1998)。

Y.Nishiura: "A hidden bifurcational structure for self-replicating dynamics"ACH-Models in Chemistry. Vol.135,No.3. 343-360 (1998)

Y.Nishiura：“用于自我复制动力学的隐藏分叉结构”化学中的 ACH 模型。

H.Imai: "Numerical simulation of free boundary problems in quadruple precision arithmetic using explicit schemes"数理解析研究所講究録,京都大学. Vol.989. 18-30 (1997)

H.Imai：“使用显式方案的四倍精度算术中的自由边界问题的数值模拟”京都大学数学科学研究所第 989 卷（1997 年）。

Tarmizi (Coauthor : T. Takeuchi, H. Imai): "NUMERICAL SIMULATION OF ONE-DIMENSIONAL FREE BOUNDARY PROBLEMS IN INFINITE PRECISION"Advances in Mathematical Sciences and Applications. Gakkotosho. Vol. 10, No. 2(to appear). (2000)

Tarmizi（合著者：T. Takeuchi、H. Imai）：“无限精度一维自由边界问题的数值模拟”数学科学与应用的进展。

H. Imai: "On numerical computation of Lyapunov exponents of attractors in free boundary problems"NIFS-PROC. (to appear).

H. Imai：“关于自由边界问题中吸引子的李雅普诺夫指数的数值计算”NIFS-PROC。