以李群DSO(n)法求解非線性微分方程式

Solving Nonlinear Ordinary Differential Equations by the Lie-group DSO(n) Methods

謝廣興(Kuang-Hsing Hsieh)

指導教授 ： 劉進賢

國立臺灣大學/工學院/土木工程學研究所/碩士(2013年)

https://doi.org/10.6342/NTU.2013.02949

摘要

模擬工程問題的數學模型大多可以常微分方程式表現，以數值分析的方式求得其解的近似值已行之有年。近年學者們研究發現常微分方程式系統經過某些代數處理後，會具有一些特殊的代數以及幾何構造。若一個演算法能夠在求解微分方程式的過程中保持在李群 結構上，則此演算法能夠使數值解具有良好的準確度及穩定性。本文發展出建立在保群算法觀念上的新方法─李群 演算法，即可保持上述構造；保群算法經過多年的發展，在諸多算例中都顯示其具有優異的數值表現，然而李群 演算法目前較少數值算例驗證其性能，是故本篇論文提供六個數值算例以MATLAB程式語言建構李群 演算法，並將其與保群算法、四階龍格-庫塔法等演算法做比較，分析歸納出新方法的優點，並指出往後的研究發展方向。

關鍵字

保群算法(GPS) ； 李群 ； 龍格-庫塔法 ； 非線性動態系統 ； 常微分方程式

並列摘要

We can model most of the engineering problems by using ordinary differential equations(ODEs), and then we often employ numerical methods to solve those equations for the approximate solutions. Studies have shown that there are some special algebraic and geometrical structures of the ODEs. If the Lie group can be preserved while solving the ODEs by some algorithm, then we will get more accurate and stable numerical solutions. This thesis constructs a brand-new algorithm─Lie group for solving non-linear ODEs based on the concept of group preserving schemes (GPS), which can preserve those structures mentioned above. The GPS have been developed for one decade,and a lot of numerical examples have shown that GPS performs well for solving ODEs, while it is deficient in the examples and evidences to show that the new method─Lie group performs well, too. Thus we apply six numerical examples, and writing the code in the programming language of MATLAB to construct Lie group algorithm. Then we compare the Lie group with the GPS, and Runge-Kutta Method of order four(RK4). Furthermore, the results of the eaxmples will be shown, and then we are going to make some conclusions, subsequently the future work of the new method.

並列關鍵字

group preserving schemes (GPS) ； Lie group ； Runge-Kutta Method ； non-linear dynamical system ； ordinary differential equations

參考文獻

37. Liu, C.S., Chang, C.W., and Chang, J.R., Past cone dynamics and backward group preserving schemes for backward heat conduction problems. CMES-Computer Modeling in Engineering & Sciences, 2006. 12(1): pp. 67-81.

1. Liu, C.S., Cone of non-linear dynamical system and group preserving schemes. International Journal of Non-Linear Mechanics, 2001. 36(7): pp. 1047-1068.

2. Chen, Y.W., Liu, C.S., and Chang, J.R., A chaos detectable and time step-size adaptive numerical scheme for nonlinear dynamical systems. Journal of Sound and Vibration, 2006. 299(4-5): pp. 977-989.

3. Zhang, S.Y. and Deng, Z.C., A simple and efficient fourth-order approximation solution for nonlinear dynamic system. Mechanics Research Communications, 2004. 31(2): pp. 221-228.

4. Liu, C.S. and Atluri, S.N., A novel time integration method for solving a large system of non-linear algebraic equations. CMES-Computer Modeling in Engineering & Sciences, 2008. 31(2): pp. 71-83.