2003 Fiscal Year Final Research Report Summary
Domain Decomposition Method for fee Boundary Problems and Its Applications
| Project/Area Number |
13640119
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | The University of Tokushima |
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Principal Investigator |
TAKEUCHI Toshiki The University of Tokushima, Faculty of Engineering, Professor, 工学部, 教授 (30264964)
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| Co-Investigator(Kenkyū-buntansha) |
IMAI Hitoshi The University of Tokushima, Faculty of Engineering, Professor, 工学部, 教授 (80203298)
NAKAMURA Masaaki Nihon University, College of Science and Technology, Professor, 理工学部, 教授 (00017419)
IKEDA Tsutomu Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (50151296)
SAKAGUCHI Hideo The University of Tokushima, Faculty of Engineering, Assistant, 工学部, 助手 (80274265)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Free Boundary / Numerical Computation / Spectral Collocation Method / DDM / High Precision / Mapping |
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| Research Abstract |
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 together
Otherwise, 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 results
It is Solved by IPNS for the free boundary problem. Numerical results are very satisfactory
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