本文主要研究投影法中的高階緊緻有限差分在非交錯網格下的空間離散方法求解非靜態不可壓縮的納維爾-史托克方程組。首先我們給定時間變數的半離散的格式，再經由投影技巧將該問題分割成兩個子問題，其中我們先求解反應-擴散方程從而得到一個中間速度場，接著將它投影到零散度空間進而取得下一個時間步的速度及壓力。對於子問題的空間變數離散，我們使用四階的緊緻有限差分法在非交錯網格下求其差分近似解。我們提供兩個數值實例來驗證此方法的精確度，包含具有正確解的流場問題和凹槽驅動流場問題。經由數值實驗結果觀察，我們確認此種高階投影法能夠取得合理的精確度。

In this thesis, we study the high-order compact difference schemes for spatial discretization on non-staggered grids in the projection methods for solving the unsteady incompressible Navier-Stokes equations. We first give a semi-discretization in time for the transient problem and then split the semi-discrete formulation into sub-problems by using the projection techniques, where we solve a reaction-diffusion equation to yield the intermediate velocity field and then project it onto the space of divergence-free vector fields to obtain the velocity and pressure at next time level. For the treatment of the spatial discretization of the sub-problems arising in the projection methods, we employ the high-order compact difference schemes of fourth-order accuracy on non-staggered grids. Two numerical examples are provided to demonstrate the accuracy of the proposed high-order projection methods, including the exact forced flow problem and the lid driven cavity flow problem. From the numerical results, we may observe that the proposed high-order projection methods can achieve a reasonable accuracy.

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