在求解大型稀疏矩陣線性系統問題上，工程師及數學家一直在尋找一效率高的演算方法。傳統上所謂的單層演算方法，隨著分析問題日益增加的計算量，慢慢達到它的計算極限。代數多重網格法是近年來快速興起的多層演算方法，延續幾何多重網格法所使用的平滑化及粗網格修正，而建立在代數化的系統方程上。其優點在於它可直接被應用於分析的問題上，而不需要任何幾何上的資訊，便可具極佳的數值強健性及延展性。
本研究將建立一套以共位體心有限體積法為基礎之數值方法，討論在求解具移動邊界之多流體不可壓縮流場中，將遭遇之速度壓力場計算、溫度場計算及波前移動位置計算。針對不同數值問題之測試，應用代數多重網格法於離散後之線性方程組求解上，結果顯示出以往單層疊代法難以求解之問題，至少可達到40倍以上之加速性；而一般性之問題求解，亦可達到近5倍之加速性，展示出代數多重網格法之問題適用性及極佳演算延展性。

In asking solving large-scale sparse matrix linear system problems, the engineer and mathematician keep looking for a method of making calculations with high efficiency. While analysis problems posed on increasingly larger computations, traditionally so-called single level methods reach its calculation limit gradually. Algebraic multigrid (AMG) is one of the multi level methods currently undergoing resurgence in popularity. The main idea behind AMG is to extend the classical ideas of geometric multigrid (smoothing and coarse-grid correction) to certain classes of algebraic systems of equations. The main practical advantage of AMG is that it can directly be applied on problem without any geometric background and poses excellent numerical robustness and scalability.
This work build up a set of numerical method based on collocated finite volume method to discuss the computation of velocity-pressure, temperature and the moving front position while solving multi-fluid moving boundary in incompressible flow simulation. In accordance with different numerical problem tests, we apply AMG method on discretized linear system solving. The results illustrate for the problems that single level methods are hard to solve, AMG can speed up at least 40 times; for the general problems, AMG also can accelerate almost 5 times. This demonstrates the range of AMG applicability and algorithmic scalability.

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