在計算流體力學的應用中，通常使用有限體積法(Finite Volume Method)配合顯式(Explicit)求解問題-利用上一步的物理性質求解通量，或隱式(Implicit)-同時求得物理性質以及通量，本研究利用方向解耦靜態直接模擬法(direction decoupled Quiet Direction Simulation)配合隱式以及較高的庫倫數(Courant–Friedrichs–Lewy number,CFL)。
本研究中，統御方程式(governing equation)為非線性，而傳統上利用隱式求解則需要化為線性方程式，因此應用牛頓-拉弗森法(Newton-Raphson method)解決非線性化問題，配合平衡通量法(Equilibrium Flux Method)或靜態直接模擬法(Quiet Direct Simulation)求解隱式之問題。牛頓-拉弗森法中需要計算雅可比反矩陣(inverse Jacobian matrix )與牛頓殘值(Newton Residual)的乘積，研究中利用有限差分法(Finite Difference Method)求解雅可比反矩陣，並用雙共軛梯度法(Biconjugate Gradient method)解出其乘積值，由於在雅可比矩陣中有許多零元素，這些不但大幅拖垮計算時間，更造成浪費記憶體的情況，因此在這裡引入壓縮稀疏矩陣(Compressed Sparse Matrix)。
另外，證明研究的可行性，將上述的方法應用在下列問題中:一維問題中-震波管的情形，二維問題包含-尤拉四震波的交互作用情形、尤拉四面波交互作用情形、爆炸波以及超音速流過階梯之情形，並使用開放式多處理(Open Ｍulti-Processing)，探討其平行化之效能。最後結論為1.隱式的庫倫數可以大於顯式，由於使用一階時間離散的關係，在隱式中較大庫倫數在精度上將會降低。2.在使用開放式多處理的情況下，開啟16線程(thread)比單一串行(serial thread)計算快上13.6倍左右的效能。3.在隱式計算上，應用壓縮稀疏矩陣能有驚人的效益和平行化的能力。此研究測試中央處理器為Genuine Intel(R) CPU @ 2.60GHz。

The application of Computational Fluid Dynamics (CFD) is an important part of any engineering analysis, with the Finite Volume Method (FVM) a key tool in modern CFD. Modern FVM simulations can be categorized into explicit simulations – where fluxes for the next time step are computed based on older values – or implicit simulations, where the new state and new fluxes across cell interfaces are computed simultaneously. This paper focuses on the application of the direction decoupled Quiet Direct Simulation (QDS) approach to implicit calculation, allowing the use of larger CFL numbers without resulting in instabilities. The governing equations are non-linear – conventional attempts at implementing an implicit approach norming involve the linearization of the governing equations. The QDS approach is fundamentally non-linear, hence this research centers on the solution of the non-linear equations. Hence, the Newton Raphson approach is employed to solve the non-linear equations for both the Equilibrium Flux Method (EFM) and the Quiet Direct Simulation (QDS). The Newton Raphson approach requires the inverse of the Jacobian of the residual functions. The Jacobian matrix is computed using a simple Finite Difference approach, while the Bi-Conjugate Stabilised (BiCGStab) method is employed for finding the Jacobian inverse. The Jacobian matrix has a considerable amount of non-zero elements, which not only influences computation time but also wastes a large amount of memory. To overcome this obstacle we employ the Compressed Sparse Matrix (CSR) storage technique.
In order to validate the implicit implementation of the QDS solver, we discuss the one-dimensional shock tube problem, a two-dimensional four contact interaction, the two dimensional four shock interaction and two dimensional blast wave problem, in addition to a hypersonic flow over a forward facing step. After comparison of the results against the EFM and existing results, we describe the parallel performance of the OpenMP implementation. The overall conclusions for the study can by summarized by (1) The allowable CFL number is greater for the implicit implementation than its equivalent explicit implementation – however, using a larger CFL number demonstrates the lower accuracy due to the use of a first order time discretization, (2) The speedup obtained using OpenMP with 16 cores is approximately 13.6x that of a single core, and demonstrating a high degree of parallel efficiency, (3) the application of CSR is critical to the efficient application and parallelization of the implicit solution.

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