1990 Fiscal Year Final Research Report Summary
Numerical Analysis of Partial Differential Equations with Free Boundaries
| Project/Area Number |
01540169
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of Tsukuba |
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Principal Investigator |
NATORI Makoto Institute of Information Sciences, Univ. of Tsukuba, Professor, 電子情報工学系, 教授 (70013745)
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| Co-Investigator(Kenkyū-buntansha) |
MIYAMOTO Sadaaki Faculty of Eng., Univ. of Tokushima, Professor, 工学部, 教授 (60143179)
IMAI Hitoshi Inst. of Info. Sci., Univ. of Tsukuba, Assis. Prof., 電子情報工学系, 講師 (80203298)
INAGAKI Toshiyuki Inst. of Info. Sci., Univ. of Tsukuba, Assoc. Prof., 電子情報工学系, 助教授 (60134219)
OYANAGI Yoshio Inst. of Info. Sci., Univ. of Tsukuba, Professor, 電子情報工学系, 教授 (60011673)
IKEBE Yasuhiko Inst. of Info. Sci., Univ. of Tsukuba, Professor, 電子情報工学系, 教授 (10114034)
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| Project Period (FY) |
1989 – 1990
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| Keywords | Free boundary problem / Partial differential equation / Equilibrium plasma / Bifurcation phenomena / Drop formation / Hodograph transformation / Boundary fit method / MAC method |
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| Research Abstract |
In this research, numerical methods to solve partial differential equations with free boundaries were investigated.
(1) Domidov developed a method using a hodograph transformation to obtain shapes of equilibrium plasma in the vessel. This method is valid only when the vessel has the polygonal boundary. We proposed an approximate resolution to obtain the shapes of plasma for the vessels which have not the polygonal boundary. The characteristics of our method is to use the conformal mapping. The validity of our method is numerically shown.
(2) Bifurcation phenomena in the free boundary problem of equilibrium plasma have been studied by Domidov and Imai and Kawarada, for one-component plasma shapes. We considered two-component plasma and bifurcation of both one and two component plasma shapes is analyzed numerically. Numerical results show that in spite of the simplicity of the problem, many types of bifurcation occur.
(3) We developed a method for finding bifurcation points along solution curves in free boundary problems. In this method, a point along a solution curve is determined as a bifurcation point where the smallest eigenvalue of a linearized problem is equal to zero. In order to verify the proposed method, numerical computations are carried out.
(4) We proposed a numerical method which is useful for simulating flows in drop formation from a capillary. In our method, the boundary fit method is used to solve the Navier-Stokes equation with a free boundary. At the same time, a new method on the idea of the MAC method is proposed in the case of the splitting of free boundaries when deformation is large.
(5) We developed a new method for the reorthogonalization in the Lanczos algorithm.
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