為了求解Euler(尤拉)方程式所統御之流動現象，本研究中提出一新方法根據有限體積法(FVM)所求解，稱為結合平衡通量法(Combined Equilibrium Flux Method, CEFM)。其包含了平衡通量法(Equilibrium Flux Method, EFM)和平衡介面法(Equilibrium Interface Method, EIM)。結合平衡通量法(CEFM)利用薩瑟蘭方法(Sutherland's law)計算鬆弛係數(relaxation term),並用來提高EFM的精準度以及EIM的穩定性. 此研究中將此方法進一步的拓展到二階空間上的精準度，利用通量限制器(flux limiter) 避免空間上所產生的非物理震盪。同時運用牛頓-拉弗森方法(Newton-Raphson method)將熱完全氣體(Thermally perfect gas)在模擬中測試。
由於在計算流體力學(CFD)模擬當中，為了獲得更精確的結果，所需耗費的計算成本也隨之增加。本研究利用多重統一計算架構(CUDA)與輝達(Nvidia)的圖型處理器(GPU)建立平行計算的架構，達到利用平行運算降低計算成本的目的。為了驗證此方法的可行性以及平行計算的效率，本研究利用基準問題來做測試，像是一維震波管，二維爆炸波，二維尤拉四面波交互作用，高速流過障礙物和階梯問題等。本研究所採用的設備為單核心英特爾i5-4690以及圖形處理器GTX Titan Black.

In order to solving the governing equation such as Euler and Navier-Stokes equations for non-ideal (real) gases, we propose a method, namely the “CEFM (combined equilibrium flux method)” which combines two existing equilibrium methods, specifically EFM (equilibrium flux method) and EIM (equilibrium interface method). The core of the method lies in the relaxation from one method to the other using a local relaxation time. In this study, we use Sutherland's law to calculate the viscosity and according relaxation, which then provides the interface flux for each cell surface. Through this approach, we demonstrate that we improve the accuracy the results over the EFM method while also demonstrating increased robustness over the EIM sachem. To further increase accuracy, we reduce spatial truncation error through the extension of CEFM method to second order spatial accuracy using a Total Variable Diminishing (TVD) approach. Accordingly, flux limiters are employed to avoid non-physical space over the space. Furthermore, the solver models real gases - specifically, thermally perfect gas - through the use of non-linear specific heat capacities, solving temperature from energy using the Newton-Raphson method.
For the sake of obtaining a precise solution, such methods often require large computational cost, and Computational Fluid Dynamics (CFD) approaches such as these are no exception. Hence, we utilize Nvidia’s Graphics Processing (GPU) and CUDA to reduce the computational time through the implementation of a parallel explicit solver. In order to verify the validity of CEFM method and the parallel efficiency, several benchmarks including the one-dimensional shock tube problem, two-dimensional blast wave, flow past bluff body, Euler four contact and Forward Facing Step Problem are tested. The results show that the method provides a suitable balance between the EFM and EIM methods while obtaining a speedup (on average) of approximately 1000x that of a single CPU core when using an Intel i5-4690 and a GTX Titan Black.

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