臺灣博碩士論文加值系統

我們以FFT為基礎發展了一個簡單、有效率的快速解法來解圓盤上的雙調和方程。雙調和方程可被視為一對連結的調和方程。首先，我們利用截形的傅立葉級數得到一組連結的ODE方程，然後用二階的中間差分法來解。利用縮半格網格點的方法，我們避免了在原點所需的邊界條件。接著利用Sherman-Morrison公式來解所推得的線性系統。在MxN個網格點下，整體所需的計算量為 。最後，我們提出了數值收斂性和一些原盤上不可壓縮的Navier-Stokes方程的應用。

We develop a simple and efficient FFT-based fast direct solver for the biharmonic equation on a disk. The biharmonic equation is regarded as a coupled system of harmonic problems. We first use the truncated Fourier series expansion to derive a set of coupled singular ODEs, then we solve those singular equations by a second-order finite difference discretization. Using a radial grid with shifting a half mesh away from the origin, we can handle the coordinate singularity easily without pole conditions. The Sherman-Morrison formula is then applied to solve the resultant linear system in a cost-efficient way. The computational complexity of the method consists of
O(MN log2 N) arithmetic operations for M xN grid points. The numerical accuracy check and some applications to the incompressible Navier-Stokes flows inside a disk are conducted.

1 Introduction
2 Fast direct solver for the biharmonic equation on a disk
2.1 The biharmonic equation
2.2 Truncated Fourier series expansion
2.3 Spatial discretization
2.4 Boundary conditions
3 Navier-Stokes equations in the vorticity-stream formulation
3.1 Incompressible Navier-Stokes equations
3.2 Time integration scheme
3.3 Spatial discretization and Boundary conditions
4 Numerical results
4.1 Accuracy check for the biharmonic equation
4.2 Accuracy check for the Navier-Stokes equations
4.3 Moving-wall problem
4.4 Tripole formation
5 Conclusion

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