以基本解法求解赫姆霍茲、擴散及柏格斯方程式

Abstract

本論文主要在探討和應用無網格基本解法求解赫姆霍茲、擴散及柏格斯方程式。首先是波導管問題之赫姆霍茲方程式。同邊界元素法僅需邊界點和源點，但無需點與點間的關係和數值積分，即可以基本解法結合奇異值分解法求得波導管的截止波數，進而求得模態圖，成功模擬橢圓波導管問題，此法可有效省下記憶體空間。其次，求解半無窮域之史托克斯第一問題及第二問題。僅有唯一受制邊界條件，基本解法無受限於計算域形式，可成功求解半無窮域和外域問題。以基本解法求解非穩態問題，直接應用具時間項的基本解，不需拉普拉斯轉換也不需要時間微分項的離散，可簡化程式，加速運算時間。以往基本解法多應用於線性問題上，本論文成功應用基本解法求解非線性的柏格斯方程式。文中分別以兩個方法將非線性柏格斯方程式轉換成線性擴散方程式，進而以基本解法求解擴散方程式後，由逆轉換求得柏格斯方程式的解。其一為尤拉-拉格朗日法，其二為柯爾霍普夫轉換。此二法均經由在空間-時間域中擺放源點，便可求解出不斷隨著時間變化的解直達穩態。由於各個數值實驗均獲得準確結果，也符合數值的穩定性與一致性，因此無網格基本解法乃一值得研究發展的高效率計算方法。

The method of fundamental solutions (MFS) is one of the popular meshless methods, gaining attention in the recent past. Since this method is free from the integration of the singular functions, this method has been applied for the solution of partial differential equations representing many engineering problems. The present thesis focuses on the application of the MFS to simulate problems of elliptical waveguides, Stokes’ first and second problems and Burgers’ equation. Initially the MFS was utilized to solve elliptical waveguide problems by solving the Helmholtz equation using the singular value decomposition (SVD) method. The method could predict the results for the cutoff wavelengths in close agreement with analytical results. Later the MFS was applied to solve unsteady Stokes’ first and second problems. The time derivatives are handled by a time-space domain concept, which completely avoids the requirement of Laplace transformation or the finite difference scheme to discretize the time derivatives. Results obtained for the unsteady Stokes’ first and second problems indicate that the MFS could predict results closer to the analytical solutions. An error analysis carried out also demonstrates that the proposed numerical scheme based on the MFS can produce stable numerical results for unsteady problems solved on semi-infinite domain. Finally, the MFS procedure was extended to solve non-linear Burgers’ equation in combination with the Eulerian-Lagrangian method and the Cole-Hopf transformation independently. The numerical experiments demonstrate that the MFS performs very well in combination with the above schemes to solve non-linear partial differential equations as well. Results obtained for many test cases of the non-linear Burgers’ equations in 1-D and 2-D domains indicate the present scheme could produce results closer to the analytical results. The results discussed in the thesis show that the MFS is a powerful meshless numerical scheme to solve non-linear partial differential equations.