以基本解方法求解對流-擴散、柏格斯及奈維爾-史托克斯方程式

Abstract

在本論文中，採用尤拉-拉格朗日基本解方法分析對流-擴散方程式、柏格斯方程式以及奈維爾-史托克斯方程式。尤拉-拉格朗日基本解方法是由尤拉-拉格朗日法以及基本解方法所組成，因此兼具兩種方法的優點。利用尤拉-拉格朗日法將具有對流項之非線性偏微分方程式轉換成線性偏微分方程式，接著再以基本解方法求解轉換後之線性偏微分方程式。在基本解方法中，將數值解表示為非穩態基本解的累加，因此可以保持無需網格以及數值積分的優點。 在開始時，將尤拉-拉格朗日基本解方法用來分析一維至三維的對流-擴散方程式，並且將數值解與解析解做比較，均可以得到非常好的結果。再來將尤拉-拉格朗日基本解方法應用在求解多維度的柏格斯方程式，在求解過程中未知變數會自然的分離，所以計算的效率會大為提高。接著將基本解方法與非穩態史托克斯例加以合併，用以求解非穩態的史托克斯方程式。將此數值方法應用在非規則形狀的計算域上，更能呈現此數值方法的種種優點。最後將尤拉-拉格朗日基本解方法與非穩態史托克斯例合併，用以求解奈維爾-史托克斯方程式。在此求解過程中只需要速度的邊界以及初始條件，可以避免傳統數值方法面臨的壓力邊界條件的問題。在本文中提出的尤拉-拉格朗日基本解方法的穩定性與一致性可以經由上述一連串數值實驗加以驗證。因此在分析對流-擴散方程式、柏格斯方程式以及奈維爾-史托克斯方程式時，所提出之尤拉-拉格朗日基本解方法，可以視為一種簡單並有效率之數值計算方法。

The Eulerian-Lagrangian method of fundamental solutions (ELMFS), which is a combination of the method of fundamental solutions (MFS) and the Eulerian-Lagrangian method (ELM), is proposed in this thesis to deal with the advection-diffusion, Burgers’ and Navier-Stokes equations. The ELM is adopted to convert the non-linear partial differential equations including the convective terms to the linear time-dependent partial differential equations and then the resultant partial differential equations are solved by the MFS based on the time-dependent fundamental solution. Initially, the proposed ELMFS is used to analyze 1D, 2D and 3D advection-diffusion equations which describe transport phenomena. The results of advection-diffusion equations are almost identical with the analytical solutions and other numerical results. Then the ELMFS is adopted to study the solutions of the multi-dimensional Burgers’ equations. During the solution procedures, the unknowns in the partial differential equations are decoupled in order to improve the efficiency of the simulation. Furthermore, the time-dependent MFS and the unsteady Stokeslets are combined together to solve the unsteady Stokes problems. The flexibility and the robustness of the MFS are demonstrated by solving the unsteady Stokes problems in irregular domains as well. Finally, the ELMFS based on the unsteady Stokeslets is adopted to analyze the Navier-Stokes equations in primitive-variable form. The stability and the consistency of the proposed ELMFS are examined in a series of numerical experiments, which demonstrate that the ELMFS can be considered as a simple and efficient numerical method in handling the nonlinear partial differential equations with convective terms.