2005 Fiscal Year Final Research Report Summary
CONSERVATION AND STABILITY OF DYNAMICAL SYSTEMS AND ITS NUMERICAL ANAKYSIS
| Project/Area Number |
15340029
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | NAGOYA UNIVERSITY |
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Principal Investigator |
MITUI Taketomo Nagoya University, Grad. Sch. of Information Science, Professor, 大学院・情報科学研究科, 教授 (50027380)
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| Co-Investigator(Kenkyū-buntansha) |
OKADA Masami Nagoya University, Fac. of Urban Liberal Arts, Tokyo Metropolitan Univ., Professor, 都市教養学部, 教授 (00152314)
KOTO Toshiyuki Nagoya University, Grad. Sch. of Information Science, Assoc. Professor, 大学院・情報科学研究科, 助教授 (30234793)
SAKAJO Takashi Hokkaido Univ., Grad. Sch. of Sci., Assoc. Professor, 大学院・理学研究科, 助教授 (10303603)
SUGIURA Hiroshi Nanzan University, Fac. of Math. Sci. and Info. Eng., Professor, 数理情報学部, 教授 (60154465)
SUGIHARA Masaaki University of Tokyo, Grad. Sch. of Info. Sci. and Tech., Professor, 大学院・情報理工学系研究科, 教授 (80154483)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Computational science / Numerical analysis / Dynamical systems / Stability / Conservative system / Discrete variable methods / Nonlinear equations / Eigenvalue analysis |
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| Research Abstract |
The project targets a study of dynamical systems from both analytical and numerical points of view with the emphasis on the conservation and the stability of the system. A typical mathematical formulation of dynamical system is attained through the Hamilton canonical form. Then its symplecticness has much significance in the theoretical as well as the numerical analysis. As such an interrelationship between analysis and numerics for the conservation property of dynamical systems has attracted many attentions of theorists and practitioners. The project was planned to carry out an in-depth study and to open new aspect of the topic. It can achieve the following items.
(1) Correspondence between continuous and discrete dynamical systems from numerical point of view
Recent development of the so-called discrete variational derivative (DVD) method sheds a new light upon the discrete variable numerical methods. DVD method can derive many finite difference methods for (nonlinear) partial differen
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tial equations to preserve its invariant quantities in a natural manner. Also a trial to extend the principle to the Galerkin-type numerical methods can attain a success.
(2) Interrelationship between numerical stability and numerical solution of conservation or dissipation type
Generally numerical stability is significant for numerical solutions of dynamical systems. It discriminates the conservation and the non-conservation of the solutions as well. As a typical example, a study of vortex sheet motion in fluid dynamics reveals the interrelationship and can give a precise diagram of the motion.
(3) Numerical analysis of dynamical system with time-delay or randomness
Many of delay differential equations and stochastic differential equations are dissipative. Then a question arises what role the numerical stability plays for these equations. Our study shows that the numerical stability is crucial for the solution, and consequently can endorse the numerical solution as simulation of the phenomena governed by such equations.
(4) Application of the study to large-scale simulations of the dynamical systems
Large-scale simulation of the system requires the stability more. Also, for a long-term integration of the system more accurate numerical schemes are called for. In some practical examples, like as eigenvalue analysis, computational fluid dynamics etc., which have practical significance, more sophisticated and powerful methods are developed. There parallel computation improves the performance much, under the condition of being stable, for numerical solution.
During the term of the project, an international conference, 2005 International Conference on Scientific Computation and Differential Equations (SciCADE05), was held in Nagoya with a good support of the project. More than 250 people, among whom around 170 were from overseas, joined the conference and exchanged new ideas each other. Most of their discussions were on the topic of the project. Less
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Research Products
(101 results)
