一維守恆加權基本不震盪法應用於明渠流模擬

常用於描述河流、湖泊、河口及近海流場之淺水波方程，在數學上屬非線性雙曲線型偏微分方程式。傳統有限差分法無法同時滿足在不連續面無數值震盪，及在光滑區域擁有高精度解之要求，常造成運算結果過份消散或產生數值震盪，且無法滿足守恆律。加權基本不震盪法（WENO）求解雙曲型方程式有二優點：對於震波等不連續面之傳遞分辨率高，且易於推廣至高維度情況及含有複雜源項之問題。本文引用守恆加權基本不震盪法，搭配有限體積法求解一維淺水波方程，並針對特徵速度求法作修正，改善該方法無法處理乾濕交界及稀疏波（rarefaction wave）問題，同時由數學離散差分式證明本文方法滿足守恆律。最後藉由水流經障礙物、潰壩、稀疏波及乾床等實驗室案例之模擬，驗證本文所提修正方法之正確性。

關鍵字

加權基本不震盪法；有限體積法；守恆律；乾床

並列摘要

Shallow water equations have been usually used to describe the flow fields in rivers, lakes, estuaries, and seashores. Shallow water equations are one of the hyperbolic-type PDE's in mathematical realm. Traditional finite difference methods cannot have high accuracy of the solution in smooth regions and without numerical oscillations at the singularities at the same time. They usually cause the simulation results having numerical damping, numerical oscillation, and not satisfying the conservation law. Solving the hyperbolic-type equations by WENO has two significant advantages：the solution can describe the phenomena clearly in the region with singularities. Besides, WENO scheme is easy to apply to two-dimensional system, three-dimensional system and even to flow fields with complicated source terms. Considering all these advantages, this paper combines conservative WENO with finite volume method for solving the one dimensional shallow water equations. With the modification of determing characteristic velocity, the original scheme can be improved to deal with the dry-bed problem and the rarefaction wave problem. Moreover, by deducing mathematical differential equations, the combination of WENO and finite volume method can be proved to satistify the conservation law. Finally, this paper demonstrates the validity of the proposed scheme by simulating several laboratory experiments, e.g., flow over a bump, dam break, rarefaction wave, and dry bed problems.

並列關鍵字

weight essentially nonoscillatory method (WENO) ； finite volume method ； conservation law ； dry bed

國際替代計量

一維守恆加權基本不震盪法應用於明渠流模擬

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