數值模式於淺水方程式與理查方程式

Numerical Models for Shallow Water Equations and Richards Equation

項建昌(Chien-Chang Hsiang)

指導教授 ： 楊德良

國立臺灣大學/工學院/土木工程學研究所/博士(2014年)

https://doi.org/10.6342/NTU.2014.00847

摘要

本論文求解了多維度淺水方程式與地下水方程式，利用數值方法來驗證控制方程式數值解之精確度、效率與實用性，目的在發展一套可用於工程問題之方便快速又精確的數值方法。所使用之數值方法主要包含徑向基底函數(RBF)之無網格法與質點追蹤有限元素法。徑向基底函數之無網格法的優勢在於可免去複雜計算區域之網格關係建立與數值積分程序，但需要建立全場計算點的聯立方程式之全矩陣求解系統，此步驟消耗大量計算資源並減低計算效率。故於本研究中，研究了局部化無網格計算方法，考慮較少數的重要參考點，以達到在降低最少計算精確度的前提下減少計算資源之消耗。質點追蹤有限元素法主要是利用質點沿著特徵線傳動的概念，可求解帶有較不連續場值的問題，但對於高度非線性如飽和與不飽和之地下水問題，直接追蹤精確度低，倘若盲目增加計算點，又影響計算效率。本研究中將方程式改寫為較有效率之平流傳動形式，並利用追蹤質點結合有限元素法以達到高計算效率與高精確度之模擬結果。結合無網格法與質點追蹤技巧，便可於複雜邊界問題如淺水方程式與理查方程式中，得到高效率高精確度的效果，是一個具有高實用性的數值模式。

關鍵字

徑向基底函數 ； 無網格法 ； 特徵線法 ； 淺水方程式 ； Richards方程式

並列摘要

In this dissertation, the multi-dimensional shallow water equations (also called the de Saint Venant equations in its one-dimensional form) and the Richards equation are considered. The motivation is to develop a convenient, efficient and accurate numerical scheme for engineering problems. The numerical modeling are used to verify the accuracy, efficiency and applicability of the numerical solutions for the above governing equations. The mainly used numerical methods include the radial basis functions (RBFs) meshless method and the mixed Lagrangian-Eulerian method with finite element method (MLE-FEM). The advantage of the RBF meshless method is to avoid the mesh generation and numerical integration in complicate domain problems. However, the full matrix system in the computing spends a lot of computational resource and time. The localized meshless method is applied in this study to avoid the full matrix solver. By considering only the important reference points, the computational cost can be reduced without losing much accuracy. By using the concept of particle tracking along the characteristic lines, the problems with discontinued field values can be solved. However, this method is not accurate in directly solving the highly non-linear problems, such as the saturated-unsaturated ground water flow problems. The computational efficiency will reduce if we increase the computational nodes blindly. In this study, the governing equations are derived into advection forms, and are solved by the MLE-FEM scheme. The simulative results are efficient and accurate by adapting computational nodes while tracking the particles. By combining the meshless methods and the particle tracking technique, the numerical methods have high applicability with high efficiency and accuracy in complicate boundary problems, such as shallow water and Richards equations.

並列關鍵字

Radial basis function ； meshless ； particle tracking ； shallow water equations ； Richards equation

參考文獻

[1.R1] Kansa E.J. (1990a) Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates. Computers Math. Applic. Vol. 19, No. 8/9, pp. 127-145.

[1.R2] Kansa E.J. (1990b) Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers Math. Applic. Vol. 19, No. 8/9, p. 147-161.

[1.R3] Shu C., Ding H., Yeo K.S. (2003) Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. Computer Methods in Applied Mechanics and Engineering, 192, pp. 941-954.

[1.R4] Tolskykh A.I., Shirobokov D.A. (2003) On using radial basis functions in a “finite difference mode” with applications to elasticity problems. Computational Mechanics, 33, pp.68-79.

[2.R1] Burguete J., Navarro P.G. (2001) Efficient construction of high-resolution TVD conservative schemes for equations with source terms: application to shallow water flows. International Journal for Numerical Methods in Fluids, 37, pp. 209-248.