臺灣博碩士論文加值系統

此論文主要目的是著重於利用多重網格法來解決具有strongly discontinuous coefficients的橢圓方程式。首先介紹如何使用多重網格法來解決三維度具有strongly discontinuous coefficients的橢圓方程式並提供一些數值測試結果，並展示一些與其他數值方法比較的數據結果。然後應用此方法於以下兩個數學模型中，其中一個是兩相不可壓縮流與不相容的水流問題，另一個是Navier-Stokes方程式。而在這兩個數學模型上，我們利用Locally conservative Eulerian-Lagrangian methods (簡稱LCELM) 來計算這兩個數學模型的transport方程式，並針對這兩個數學模型展示一些數值結果。

The primary objective of this thesis is to introduce a multigrid method to solve elliptic equation with strongly discontinuous coefficients. In the beginning, we explain how to use the multigrid method to solve a 3D elliptic equation with strongly discontinuous coefficients, and then show some numerical testing results. Also, we provide some results compared with other numerical methods to show the efficency of the mutigrid method. Furthermore, we apply the multigrid method to solve two mathematical problems, one is for the waterflooding problem and the other is the incompressible Navier-Stokes equations. A locally conservative Eulerian-Lagrangian method (briefly LCELM) is used to compute the transport part of the two models. Some numerical results for the two problems will be presented as well.
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中文摘要.............................................................i
Abstract............................................................ii
Acknowledgements ..................................................iii
Contents............................................................iv
Figures..............................................................v
1. Introduction......................................................1
2. A Multigrid Method for Nonuniform Elliptic Equation...............3
2.1 Multigrid Method..............................................3
2.2 Discretized Equation..........................................6
2.3 Prolongation Operator.........................................9
2.4 Restriction Operator.........................................13
2.5 The Poisson Equation with Strongly Discontinuous Coefficient.21
2.6 Neumann Boundary Condition...................................22
2.7 Test Examples................................................24
3. The Waterflooding Problem........................................35
3.1 Computational domain.........................................37
3.1.1 Discretization in temporal domain......................37
3.1.2 Discretization in Spatial Domain.......................40
3.2 The Pressure Equation........................................41
3.3 The Transport Equation.......................................42
3.3.1 The MMOC Procedure.....................................42
3.3.2 The LCELM Procedure....................................44
3.3.3 Diffusive Fractional Step for the Saturation...........47
4. The Incompressible Navier-Stokes Equations.......................48
5. Numerical Results................................................51
5.1 Numerical Results of the Waterflooding Problem...............51
5.2 Numerical Results of the Incompressible Navier-Stokes Equations..67
6. Conclusion............................................................79
Reference.............................................................80
Appendix............................................................82

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