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題名 | 一維移流傳輸方程式之數值解法比較=A Comparison of Numerical Schemes for One-Dimensional Pure Advection |
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作者 | 許少華; 倪春發; | 書刊名 | 逢甲學報 |
卷期 | 29 1996.06[民85.06] |
頁次 | 頁133-144 |
分類號 | 443 |
關鍵詞 | 移流; 計算; 數值方法; Advection; Computation; Numerical scheme; |
語文 | 中文(Chinese) |
中文摘要 | 污染物傳輸方程式的移流項乃一階雙曲線型的偏微分方程式。若使用一階精度的 數值差分法模擬,則有嚴重的數值擴散誤差。若使用二階精度的數值方式則常有數值振盪現 象。 僅有特性法 (Characteristic Method) 因理論基礎較接近物理現象而能捕捉住移流項 的特性。本文以 Holly-Preissmann 特性線法 (1977) 為基礎,於空間軸採三次曲線內插法 應用於可蘭數小於 1.0 時,而以時間軸的三次內插於大於 1.0 之狀況。於高斯點源與連續 源分布之測試下, 本文所採用之改進 H-P 法均得良好結果,且在可蘭數 (Cr)>1 時更顯出 其適用性。 其他用來比較的常用數值方法有 Upwind, Leap Frog, Crank-Nicolson, 以及 Finite Element 法。 Upwind 與 Leap Frog 法完全不適用, Crank-Nicolson 與 Finite Element 法有顯著的數值振盪誤差。 |
英文摘要 | Pure advection term in the well known advection-dispersion equation represents a first order P.D.E. of hyperbolic type. Numerical methods with first order accuracy, such as Upwind, always create severe numberical diffusion. For schemes with second order accuracy, usually suffer with numerical oscillation and phase error. Characteristic methods which follow characteristic lines and transform P.D.E. problems into Ordinary Differential Equations can capture the physics of pure advectio. In this article, Holly- Preissmann's Characteristic method (1977) is adopted and refined. For Courant Number (Cr) ≦ 1, traditional Hermit Cubic Interpolating between two points on space-line is adopted. For Cr>1, interpolation on time-line is performed instead. Two kinds of benchmark problems are tested which are Gauss-Hill distribution and Step function distribution of initial plume concentration. The refined H-P scheme is superior to other schemes in both cases in terms of accuracy and stability. five selected methods including Upwind, Leap Frog, Crank-Nicolson, and Finite Element are compared and discussed. |
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