可預測分歧點及調整步長的延伸軌跡之數值方法

Abstract

在此篇論文中，介紹了描繪參數相關的非線性方程式之解軌跡的虛擬弧長延續數值法。這樣的問題出現在很多應用中，例如：初值問題的定態解及兩點邊界值的問題等等。我們藉由數值實驗顯示了與一階Lagrange矩陣內插為根據之分歧點預測計劃結合的虛擬弧長延續過程可以預測出單純分歧點的位置及利用簡單的子空間投影策略可描繪出次要的分歧軌跡。此外，由Seydel提出，建立在打靶法的觀念上之步長控制技巧可以大幅改善計算效率。我們也舉例、以圖表說明，呈現一些數值實驗結果。

In this thesis, we investigate the well-known pseudo-arclength continuation method for numerically tracing the solution paths of parameter-dependent nonlinear equations. Such problems arise in many applications, e.g., steady-state solution of initial-value problems and two-point boundary value problems, etc. We show by numerical experiments that the pseudo-arclength continuation procedure incorporated with bifurcation prediction scheme based on first order Lagrange matrix interpolation can detect the locations of simple bifurcation points and trace the secondary bifurcation paths with simple subspace projection strategy. Furthermore, a step-control technique based on the idea of shooting method proposed by Seydel can greatly improve the efficiency of the computational effort. A few numerical experiments and results are presented for illustration.