混合通量分裂式有限體積算則在二維淺水波方程式之研發

Developments of the Hybrid Flux-Splitting Finite-Volume Schemes for 2D Shallow Water Equations

郭文達(Wen-Dar Guo)

指導教授 ： 林國峰

國立臺灣大學/工學院/土木工程學研究所/博士(2004年)

https://doi.org/10.6342/NTU.2004.00808

摘要

發展一個能精確無虛假震盪地解析不連續水流現象且不會產生過多的數值耗散之算則，ㄧ直是計算淺水動力學上重要的課題。本研究將近年來於計算空氣動力學中所廣泛採用之數值概念，應用到淺水動力學之數值計算，進而研發三個求解二維淺水波方程式之混合通量分裂式有限體積算則。
首先，採用有限體積法離散控制方程式，並且採用對流逆風分裂法(AUSM)進行數值通量之估計，進而發展出ㄧ個無需Jacobian矩陣之混合通量分裂運算式。接著，以此運算式為基礎，可將有限體積基本運算式進ㄧ步地離散為具有一階準確度之混合通量分裂式有限體積(HFS)算則。所發展之算則因無需Jacobian矩陣之運算，可節省釵h計算時間，且於撰寫程式方面，顯得相當容易。最後，本研究分別採用兩種不同之二階精度延伸途徑，包括單調守恆律法(MUSCL)搭配非線性坡度因子之途徑以及預測、校正二步逐分量之全變量消逝(TVD)途徑，以提高一階算則對於不連續解之捕捉解析能力。
為了驗證本文所研發算則對於水躍與震波等不連續水流之解析能力，進行釵h淺水流動問題之數值模擬，包括ㄧ維理想化之溃壩、極端膨脹波、定量明渠跨臨界流以及斜水躍等問題；同時將模擬結果與解析解以及文獻上其它發展完全之震波捕捉逆風算則之模擬結果進行各種數值性能之比較。經由模擬結果顯示，即使在具有乾床情形、底床坡度與摩擦項存在時，本文之算則仍能夠正確地捕捉不連續水流現象且不產生數值震盪。另外，對於計算含特徵值零點之膨脹波不會產生違反熵的解。由模擬結果可進一步地證明本文算則之另ㄧ個優點：具有同時結合通量向量分裂(FVS)算則之數值效益與通量差分裂(FDS)算則之數值精度。
最後，將本文發展之算則模擬應用於二維潰壩水流問題。經由模擬結果顯示，本文之算則能夠正確地模擬出膨脹波、震波、反射震波、回流以及乾濕交界波前等水流現象。

關鍵字

混合通量分裂式有限體積算則 ； 淺水波方程式 ； 有限體積法 ； 對流逆風分裂法 ； 全變量消逝

並列摘要

The development of a numerical scheme that resolves sharp discontinuities without spurious oscillations and do not produce too much numerical dissipation is of great importance in the computational shallow-water hydrodynamics. In this thesis, three hybrid flux-splitting finite-volume schemes are proposed for solving two-dimensional shallow water equations.
In the framework of the finite volume method, a hybrid flux-splitting algorithm without Jacobian matrix operation is established by applying the advection upstream splitting method (AUSM) to estimate the cell-interface fluxes. Based on the proposed algorithm, a first-order hybrid flux-splitting finite-volume (HFS) scheme is developed, which is robust and rather simple to implement. To improve the numerical resolutions of discontinuities, the monotonic upstream schemes for conservation laws (MUSCL) method with limiters and the two-step component-wise total variation diminishing (TVD) method are adopted for the second-order extensions.
The proposed three finite-volume schemes are verified through the simulations of the 1D idealized dam-break, extreme rarefaction wave, steady transcritical flow and oblique hydraulic jump problems. The numerical results by the proposed schemes are compared with those by other shock-capturing upwind schemes as well as exact solutions. It is demonstrated that the proposed schemes are accurate and efficient to capture the discontinuous solutions without any spurious oscillations in the complex flow domains with dry-bed situation, bottom slope or friction. In addition, the proposed schemes are proven to produce no entropy-violating solution and to achieve the benefits combining the efficiency of flux-vector splitting (FVS) scheme and the accuracy of flux-difference splitting (FDS) scheme.
Furthermore, the proposed schemes are applied to simulate several 2D dam-break problems, including the partial dam breaking, circular dam breaking and four experimental dam-break problems. The simulated results show that the proposed schemes can deal with the rarefaction waves, shocks, the reflected shocks, the reverse flows and the dry/wet fronts very well.

並列關鍵字

Finite volume method ； Advection upstream splitting method ； Hybrid flux-splitting finite-volume schemes ； Shallow water equations ； Total variation diminishing

參考文獻

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