非線性孤立波之高階數值解析

Abstract

本文將原本卡氏座標下之孤立波流場映射至 為半圓形區域成一個沉流與一個源流之偶流(Doublet)，流場複變流速勢之特性，依複變運算、級數展開及疊代法概念，首先推導出一個無窮tanh型級數之孤立波解，此與McCowan (1891) 所提出孤立波之完全解形式相同。 本文特別將此tanh型級數之複變形式展開極座標形式，克服tanh函數冪次方展開項數過多之缺點，並以配置法依邊界條件使用牛頓疊代法求取係數。討論配置法條件上之選定，使用Wu et al.(2005a)之波高條件計算，由小孤立波至大孤立波，分別計算動力特性，由測試後本文建議數值方法之計算技巧條件為垂直切割方法、擷取階數為96階、初始值為已計算相近波高條件答案進行疊代為最佳。本文最佳結果與Wu et al.(2005a)比較，動力特性相對誤差精度皆在2.5%以下，而所有計算之動力特性由小孤立波至大孤立波皆符合動能大於位能，證實本文結果能精準的描述由小孤立波至大孤立波之動力特性。

The paper is to propose a numerical method to calculate highly nonlinear solitary waves. The key expression of solitary waves is first derived in the paper. Through conformal mapping the whole flow in the -plane is mapped into a semicircular region in the -plane forming a doublet that includes a source and a sink of equal strength. The complex velocity potential for the doublet in the -plane is transformed by a series using a condition of analytical function for the whole field except the doublet and iterative approximation. The deviated infinite tanh-type series for solitary waves is identified by the McCowan’s solution. The deviated infinite tanh-type series for solitary waves is tedious to expand the complex form into the real part and imaginary part so that its alternative polar form is applied for propitiously avoiding the disadvantage of series expansion. The collocation method and Newton iterative method combined by 96 terms truncated from the infinite series and vertical equal-space cutting are examined key numerical algorithms in the proposed calculation. For dwarf solitary wave to high solitary waves all calculated dynamic properties are close to those of Wu et al. (2005a) by small relative errors less than 2.5%. The resulting comparison verifies the validity of the proposed numerical calculation of highly nonlinear solitary waves.