摘要
本論文主要的研究方向是求解不可壓縮流Navier-Stokes及磁場方程式。論文架構在多維正交曲線座標系統上，于非交錯式網格上壓力與速度耦合配置方式下，採用有限差分方法離散統御方程式，發展高階準確單調多維對流、擴散及反應項次數值算則，以期準確的求解流體力學與電磁學程式。 本研究的主要貢獻是依據質量的守?琠吽B對流項次離散算則的適用性、邊界條件設定的正確性與避免非物理性震盪的條件下，率先採用控制方程式之精確解即指數型通解作為離散模型，以非交錯式網格配置的有限差分方法，建構三維正交曲線座標系統的統御方程式。並由不可壓縮拘束條件及對流優勢所導致之數值不穩定。為了消除由於對流優勢所引致的不穩定現象，吾人嘗試建構了緊致格式及保有頻散關係格式之有限差分方法，藉由具實解之二維Navier-Stokes方程式之測試，可驗證此方法具可行性。另採用M矩陣觀念，使離散方程組在某一特定範圍下保有正定性質，使得所求解的物理量具單調性，這有助于補捉在高雷諾數流場中具不連續解特性之物理現象。其次，針對多維暫態問題，率先發展適用於提高時間上求解的準確性，分別地引入高階多步方法、顯性與隱性混合四階的方法。再則，本研究率先發展了適用于多維求解速度與壓力耦合方程組于非交錯式網格，並提出一套革新求解不可壓縮流Navier-Stokes方程組的方法。最後，本研究率先發展出適用于多維空間上處理因非線性項所造成解的震盪與不收斂性，分別建構了線性化方法與多層次方法，有效地加快于求解不可壓縮流Navier-Stokes方程組的收斂速度。在求解三維問題時往往需耗費極大的計算資源，因而使用一快速且穩定之求解矩陣方程組及計算域切割之方法為解決三維問題之首要課題。傳統之直接求解法需佔用大量記憶資源，因而有效之疊代法應運而生，PGMRES 疊代法是針對非對稱且非正定矩陣方程組之有效求解方法，除使用較小的記憶資源外，又可迅速的求解。
並列摘要
To preserve incompressibility constraint condition and avoid oscillations in cases of dominated convection, the major focal point of this dissertation is to develop some effective models for solving magnetic induction equations and the incompressible Navier-Stokes equations. In the simulation, there is one major source of instability due to an inappropriate storage of the velocity and pressure fields. This type of instabilities usually appears as oscillations seen primarily in the pressure filed. The other instability source is due to the presence of advection terms in the equations, which can result in spurious oscillations in the velocity field. These two challenging instability problems form are core of the present study. To solve the CD (convection-diffusion) model equation the FDM (finite-difference method) for the field equations under current investigation is developed on non-staggered grids. The unconditional mass conservation, the approximation of discrete convection terms, the acceleration of convergence are considered in the proposed method. The FDM is employed along with the compact scheme and the DRP (dispersion-relation-preserving) theory to overcome the convective instability that arises from the convection term for problems considered at high Reynolds numbers has been employed to enhance stability. A novel method in non-staggered grids to accelerate the nonlinear convergence of incompressible Navier-Stokes equation is also proposed. To validate these proposed methods, several 2D and 3D test cases were performed. The simulated results show that the proposed methods are highly reliable and applicable to a wide range of flow conditions. For unsteady problems, a high order multiple time-stepping scheme, which increase the time accuracy, is introduced. Linearization and multi-level methods were proposed for effectively accelerating the non-convergence arising from the convection term.