臺灣博碩士論文加值系統

當庫倫數(Courant Number)接近或大於1時，時間軸cubic spline內插法產生極大的計算誤差，這可能是使用自然端點限制之結果。本研究進行不同端點限制與特性線外延技巧對時間軸cubic spline內插特性法之影響分析，探討最佳的端點處理方法。結果顯示，非結點端點限制會造成計算不穩定，自然與二次函數端點限制，分別產生嚴重的數值擴散與數值震盪。一階導數與二階導數端點限制有相近之模擬結果，且優於自然與二次函數端點限制，但與特性線外延技巧相比較，卻有較大的均方根誤差，且產生某種程度之數值擴散與數值震盪。由此可知，端點限制嚴重影響時間軸cubic spline內插特性法之模擬結果。藉由特性線外延技巧使特性線之落點遠離端點，可以有效降低端點限制之影響。因此，時間軸cubic spline內插特性法，需配合特性線外延技巧以減少計算誤差。

The characteristics method with time-line cubic spline interpolation induces large computational error when the Courant number is close or greater than the unity. This may be due to the use of natural endpoint constraint. To find suitable treatment of endpoint this study examines the effects of various endpoint constraints and reachout technique on the characteristics method with time-line cubic spline interpolation. The results show that the not-a-knot induces computational instability when the Courant number is greater than the unity. The natural and the quadratic endpoint constraints produce large numerical diffusion and numerical oscillation, respectively. The first derivative and the second derivative endpoint constraints have similar results and are superior to the natural and the quadratic endpoint constraints. However, the reachout technique has less root mean square error than the first derivative and the second derivative endpoint constraints which induce a certain degree of numerical diffusion and numerical oscillation. This outcome indicates that the characteristics method with time-line cubic spline interpolation is largely influenced by the endpoint constraint. When the location of characteristics curve is away from the endpoint the effects of endpoint constraint can be efficiently avoided by using the reachout technique. Therefore, the reachout technique needs to be adopted to reduce the computational error from the characteristics method with time-line cubic spline interpolation.

摘要……………………………………………………………………i
Abstract………………………………………………………………ii
誌 謝…………………………………………………………………iii
目 錄…………………………………………………………………iv
表目錄…………………………………………………………………vi
圖目錄…………………………………………………………………viii
符號說明…………………………………………………………………x
第一章緒論………………………………………………………………1
1.1研究動機與目的……………………………………………1
1.2文獻回顧……………………………………………………2
1.3研究方法與步驟……………………………………………3
1.4章節介紹……………………………………………………4
第二章三次內插特性法簡介……………………………………………5
2.1特性法簡介…………………………………………………5
2.2cubic spline內插法………………………………………6
2.3時間軸cubic spline內插特性法…………………………7
第三章時間軸三次內插特性法之評估…………………………………9
3.1端點限制之評估……………………………………………9
3.2特性線外延技巧之評估……………………………………14
3.3端點限制與特性線外延技巧之綜合評估…………………16
第四章時間軸三次內插特性法之應用…………………………………18
4.1變係數移流擴散方程式之計算……………………………18
4.2Burgers 方程式之計算……………………………………19
4.3運動波漫地流之計算………………………………………20
第五章結論與建議………………………………………………………22
5.1結論…………………………………………………………22
5.2建議…………………………………………………………23
參考文獻…………………………………………………………………24
附 錄…………………………………………………………………70

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