非線性塊狀高斯消去牛頓演算法在噴嘴流體的應用

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姓名

林新倫(Hsin-lun Lin) 查詢紙本館藏

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論文名稱

非線性塊狀高斯消去牛頓演算法在噴嘴流體的應用
(Some Newton methods with nonlinear Block Eliminations for the shocked duct flow problem.)

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摘要(中)

我們考慮噴嘴流體問題，
先利用有限差分法把問題離散得到一個大型非線性系統，
對於這個大型非線性系統我們藉由牛頓法來解數值解。
在這類的問題中，流體急速減速過程中所產生的震波對於牛頓法迭代收歛性造成影響。
本論文考慮並測試一個新的方法叫做非線性塊狀高斯消去牛頓法。在這個方法中，我們定義一個
局部問題並利用局部問題找到震波正確的位置，以改進牛頓法的收歛性與收斂速度。數值實驗結果證明，
在所需電腦運算時間上之比較，我們這個新的方法優於傳統的牛頓法。

摘要(英)

Newton type method is one of most popular methods for solving a large nonlinear
algebraic system of equations arising from the discretization of partial differential
equations with applications in science and engineering. Due to the presence of normal
shock wave the convergence rate of Newton type methods for solving the discrete nozzle
flow problem becomes very slow. In this thesis, we proposed and tested some right
nonlinear preconditioned iterative algorithm to enhance robustness of Newton’’s method
and to improve it’’s convergence rate. In this method, we define a local problem, which
is governed by the same differential equation as the global problem we try to solve
while the boundary conditions are imposed to satisfy the current global approximation
at these grid points. Such solution of the local problem is able to quickly detect the
exact location of shock wave. Finally, we show numerically that our approach is better
than some traditional Newton’’s method in terms of total CPU time.

關鍵字(中)

★ 噴嘴流體
★ 牛頓法
★ 穿音速
★ 非線性消去法

關鍵字(英)

★ Nozzle flow
★ Newton method
★ transonic
★ Nonlinear eliminations

論文目次

中文摘要..................................i
英文摘要.................................ii
致謝辭..................................iii
目錄.....................................iv
圖目錄....................................v
表目錄...................................vi
1.導論....................................1
2.噴嘴流體模型與其離散方程................3
3.牛頓法..................................7
4.非線性塊狀高斯消去牛頓演算法............9
5.數值結果與討論.........................12
5-1.參數設定...........................12
5-2.解的唯一性.........................13
5-3.子區間選取.........................16
5-4.方法比較...........................17
6.結論與未來發展.........................22
參考文獻.................................23
附錄一...................................25
附錄二...................................27
附錄三...................................28

參考文獻

[1] J.D. Anderson, JR. Fundamental of Aerodynmaics. 2nd Edition, McGraw-Hill, Inc 1991, New York.
[2] T.L. Holst. Transonic flow computations using nonlinear potential methods. NASA Ames Research Center, Moffett Field, Mail Stop T27B-1, CA 94035-1000, USA. Progress in Aerospace Sciences, Vol.36,2000,pp.1-61
[3] X.-C.Cai, D.E.Keyes, D.P.Young. A nonlinear additive schwarz preconditioned inexact newton method for shocked duct flows, Proceedings of the 13th International Conference on Domain Decomposition Methods, Oct. 9-12, 2000, France.
[4] X.-C.Cai, W.D.Gropp, D.E.Keyes, R.G.Melvin, and D.P.Young.1998, Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation, SIAM J. Sci. Comput. 19:246-265.
[5] D.P.Young, W.P.Huffman, R.G.Melvin, C.L.Hilmes, and F.T.Johnson. Nonlinear elimination in aerodynamic analysis and design optimization, in Proceedings of the First Sandia Workship on Large-scale PDE Constrained ptimization, Lecture Notes in Computational Science and Engineering, Heidelberg, Berlin, New York,2002,Springer Verlag.
[6] F.-N. Hwang, and X.-C.Cai , A combined linear and nonlinear preconditioning technique for incompressible Navier-Stokes Equations, Lecture Notes in Computer Science, 3732 (2006), pp. 313-322.
[7] P.J. Lanzkron, D.J. Rose, and J.T. Wilkers, An analysis of approximate nonlinear elimination, SIAM, Journal on Scientific Computing, 17 (1996), pp. 538-559.
[8] S. Bellavia, and S. Berrone, Globalization strategies for Newton-Krylov methods for stabilized FEM discretization of Navier-Stokes equations, Journal of Computational Physics, Vol. 226, pp 2317-2340, 2007.
[9] D. P. Young, R. G. Melvin, M. B. Bieterman, F. T. Johnson, S. S. Samant, and J. E. Bussoletti.
A locally refined rectangular grid finite element method: Application to computational fluid dynamics and computational physics, Journal of Computational Physics, 92:1--66, 1991. 17.
[10] X.-C.Cai, M. Paraschivoiu, M Sarkis, An explicit multi-model compressible flow formulation based on the full potential equation and the Euler Equations on 3D unstructured meshes, Contemporary Mathematics, 2000.

指導教授

黃楓南(Feng-Nan Hwang)

審核日期

2008-6-4

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