2008-08-012024-05-17https://scholars.lib.ntu.edu.tw/handle/123456789/688657摘要：利用數值方法求解不可壓縮 Navier-Stokes方程式時，所遭遇的主要困難是因該方程式涉及非線性的對流項，當對流項在整個控制方程式中佔有優勢時，非線性量將使得矩陣方程呈現明顯的不對稱性及病態的特徵值分佈，這將大幅地增加求解的困難度；再者，在求解Navier-Stokes方程式時，該如何處理不可壓縮性的限制條件，至今仍存在著相當的難度，這也是眾多學者所欲克服之處。有鑒於此，本文發展多重網格方法，以期降低因非線性方程式中對流項所造成在數值求解時有關收斂性的困難。 在求解大量的線性方程式時，使用直接法來求解問題時，將會涉及相當龐大的計算量，因此在求解大型的線性代數方程時，通常會使用迭代方法來求解。然而在求解龐大的問題時，使用一般的迭代方法，將會遭遇到計算的不易收斂的問題；有鑒於此，多重網格(Multi-Grid)法便被提出，以期大幅將低不易收斂的問題，並能夠大幅地降低計算量。 多重網格法在提出之後，目前已受到廣泛的使用，並對於某些問題(如Laplace equation)已能得到相當好的結果。然而對於對流－擴散方程，即Navier-Stokes中之動量方程式，一般的多重網格法的有效性就會大幅的降低，尤其是在問題的對流項效應逐漸增大之時。就多重網格的理論上來說，對於每一種方程式都應有相當良好的收斂結果，而不會有選擇性地僅對於某些問題呈現較好的效果。 因此，本研究將發展具理論內涵之prolongation運算子，此一運算子在實施多重網格法時，勢必需要建構的運算子，以期能夠讓多重網格法的收斂行為能夠有效的改善，進而讓計算時間減少。此運算子將以非期次對流－擴散方程式為基底，經由理論的推導而得到。 <br> Abstract: The main difficulties in solving incompressible Navier-Stokes equations is the encountered nonlinear advection terms, in particular, when convection dominates the diffusion terms, since the resulting matrix becomes asymmetric and ill-conditioned. The matrix asymmetry and poor eigenvalue distribution will largely increase the computational difficulty. Moreover, how to resolve the problem regarding the divergence free constraint condition still remains as one of the computational challenges in solving incompressible fluid flow equations. These two above mentioned difficulties motivate the development of a new class of Multi-Grid method for resolving the difficulty resulting from the large convection term. When solving the large-sized linear matrix equations using the direct method, we will be faced with the considerable demand on computer storage and in turn huge computing time. This motivated the use of iterative method, which however yields the convergence problem. For this reason, Multi-Grid method has been proposed to resolve the problems regarding the slow convergence and the needed expensive computing time. While Multi-Grid method has been widely applied to various equations with success, in particular, in solving the Laplace equation, its effectiveness will be greatly damaged in dealing with the convection-diffusion equations investigated under the high Peclet number condition. From the theoretical viewpoint, use of Multi-Grid method is irreverent to the equations type, we are therefore motivated to develop a prolongation operator, which is indispensable on the Multi-Grid method, with theoretical grounds. It is best hoped that this new operator construction can effectively speedup the convergence and, thus, decrease the CPU time when solving the high Reynolds number flow. The development is rooted in the inhomogeneous convection-diffusion equation with theoretical content.理論prolongation運算子多重網格法Navier-Stokes方程