在工程應用上，我們要做一些不同自然現象的數值模擬時，需要快速、可靠、準確求出定義在一個複雜幾何圖形上的時間的3D不可壓縮Navier-Stokes方程組的數值解。為了能夠得到在邊界層附近較準確的解，通常需要高解析的網格，這表示我們需要用到大規模的平行計算。雖然已經花了很多年的研究，來尋找一個合適的方法求解定義在一個很細的網格和很大範圍的Reynolds數上的Navier-Stokes方程組，這仍然是一件很困難的計算工作。這篇論文的目的，在研究一些平行scalable的演算法，來解一個經由stabilized finite element method對空間做離散，和一系列implicit ODE integrators對時間做離散後的時間的不可壓縮Navier-Stokes方程組所產生的大型稀疏非線性系統。我們的平行演算法是以Newton-Krylov-Schwarz演算法為基礎，它包含了三個部分：一個非線性和線性的solver為inexact Newton method with backtracking和一個線性的solver來解Jacobian系統為Krylov subspace method以及一個preconditioner來加速線性solver的收斂為平行的overlapping Schwarz domain decomposition method。此外，我們的平行演算法是以PETSc為撰寫工具，並且加入一些其他的前處理和後處理軟體套件。這些套件包含(1)一個Cubit和以C語言為基礎撰寫的3D unstructured finite element網格產生器(2)一個網格的分割器ParMETIS來做平行的網格分割處理(3)一個ParaView的科學的visualization來展現數值結果和處理數據分析。我們的演算法會在台灣的一些平行機器上執行，來解三維的start-up lid-driven rectangular cavity flows，並且會說明我們演算法的平行效能。我們也會將平行的流體solver應用在數值模擬一個微流體系統內的微混合器。 Various numerical simulations of physical phenomena in some engineering applications often require fast, reliable, accurate numerical solutions of unsteady 3D incompressible Navier-Stokes equations defined on a complex geometry. To resolve the details of the solution in the boundary layer region, high resolution meshes are often required, which implies the need for large-scale parallel computing. Even though years of research have been spent on finding such a suitable method for solving Navier-Stokes equations on very fine meshes for a wide range of Reynolds number, it remains a difficult computing task. The goal of this thesis is to study some parallel scalable algorithms for solving large sparse nonlinear systems of equations arising from the discretization of unsteady incompressible Navier-Stokes equations, where a stabilized finite element method and a family of implicit ODE integrators are employed for the spatial and temporal discretizations, respectively. Our parallel algorithm is based on a Newton-Krylov-Schwarz algorithm, which consists of three key components: an inexact Newton method with backtracking as the nonlinear linear solver, a Krylov subspace method as the linear solver for the Jacobian systems, and a parallel overlapping Schwarz domain decomposition method as a preconditioner to accelerate the convergence rate of the linear solver. In addition, our parallel flow solver implemented by PETSc is integrated with other pre-processing and post-processing software packages. These packages include (1) A Cubit and C language based 3D unstructured finite element mesh generator; (2) a mesh partitioner, ParMETIS for the purpose of parallel processing; (3) A ParaView based scientific visualization for displaying numerical results and conducting data analysis. We report the parallel performance of our algorithms for solving three-dimensional start-up lid-driven rectangular cavity flows, which are tested on some parallel machines in Taiwan. We also present an application of the parallel flow solver to simulate numerically micromixing in a microfluidic system.
