Domain Decomposition Method for fee Boundary Problems and Its Applications

费用边界问题的域分解方法及其应用

• 批准号: 13640119
• 负责人: TAKEUCHI Toshiki
• 金额: $ 2.24万
• 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 (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640119/
• 关键词: Free Boundary; Numerical Computation; Spectral Collocation Method; DDM; High Precision; Mapping; 数値計算; 領域分割; 高精度

It is known that there are travelling wave solutions in one-dimensional hyperbolic equations and reaction-diffusion equations. As a numerical method to capture the travelling wave solutions, we propose a numerical method for tracking the level set with the arbitrary precision. The feature of this method is that the level set is considered to be a free boundary and the original problem is transformed into a free boundary problem. Free boundary problems are boundary value problems defined on domains whose boundaries are unknown and must be determined as the solution. Many practical problems are formulated as free boundary problems. Recently, numerical methods for free boundary problems have been developed and improved. But investigation of the reliability of numerical results is not easy because of the unknown shape of the domain. So, we use the domain decomposition method and the fixed domain method togetherOtherwise, IPNS(Infinite-Precision Numerical Simulation) was developed. It consists of the spectral collocation method and multiple precision arithmetic. The spectral collocation method is used for the control of truncation errors. Multiple precision arithmetic is used for the control of rounding errors. The method is applicable to PDE systems with smooth solutions. It facilitates investigation of the reliability of numerical resultsIt is Solved by IPNS for the free boundary problem. Numerical results are very satisfactory

众所周知，一维双曲型方程和反应扩散方程存在行波解。作为一种捕获行波解的数值方法，我们提出了一种以任意精度跟踪水平集的数值方法。该方法的特点是将水平集视为自由边界，将原问题转化为自由边界问题。自由边值问题是定义在边界未知的区域上的边值问题，其边界必须确定为解。许多实际问题都可以表述为自由边界问题。近年来，自由边界问题的数值方法得到了发展和改进。但由于区域的形状未知，对数值结果的可靠性进行研究并不容易。因此，我们将区域分解法和固定区域法结合起来，并发展了无限精度数值模拟（IPNS）。它由谱配置法和多精度算法组成。谱配置法用于截断误差的控制。采用多精度算法控制舍入误差。该方法适用于具有光滑解的偏微分方程组。它便于研究数值结果的可靠性。数值结果是令人满意的

项目成果

Takao Hanada, N.Ishimura, Masa Aki Nakamura: "Numerical and analytical study on the Eguchi-Oki-Matsumura equations"Proceedings of Fifth China-Japan Seminar on Numerical Mathematics, Science Press. 38-46 (2002)

花田隆夫、N.Ishimura、Masa Aki Nakamura：《Eguchi-Oki-Matsumura方程的数值与分析研究》第五届中日数值数学研讨会论文集，科学出版社。

花田孝朗, 石村直之, 中村正彰: "Eguchi-Oki-Matsumura方程式の解の構造"京都大学数理解析研究所講究録. Vol.1288. 11-19 (2002)

Takaaki Hanada、Naoyuki Ishimura、Masaaki Nakamura：“Eguchi-Oki-Matsumura 方程解的结构”京都大学数学分析研究所 Kokyuroku。1288. 11-19 (2002)

T.Nishida, T.Ikeda, H.Yoshihara: "Pattern formation of heat convection problems, Mathematical Modeling and Numerical Simulation in Continuum Mechanics"Lecture Notes in Computational Sciences and Engineering. Vol.19. 209-218 (2002)

T.Nishida、T.Ikeda、H.Yoshihara：“热对流问题的模式形成、连续介质力学中的数学建模和数值模拟”计算科学与工程讲义。

Hitoshi Imai: "Parallel Computing in Infinite Precision Numerical Simulation for PDE Systems"Proceedings of Fifth China-Japan Seminar on Numerical Mathematics, Science Press. 141-153 (2002)

今井仁：《偏微分方程系统无限精度数值模拟中的并行计算》第五届中日数值数学研讨会论文集，科学出版社。

Hitoshi Imai: "Numerical Computation of Lyapunov Exponents Related to Attractors in a Free Boundary Problem"Nonlinear Analysis. Vol.47. 3823-3833 (2001)

Hitoshi Imai：“自由边界问题中与吸引子相关的李雅普诺夫指数的数值计算”非线性分析。