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題 名 | 以分離運算子解一維移流延散方程式之數值參數準確性分析=Acceptable Ranges of Numerical Parameters on Solving One-Dimensional Advection/Dispersion Equation by a Split-Operator Scheme |
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作 者 | 許少華; 林伯聰; 倪春發; | 書刊名 | 農業工程學報 |
卷 期 | 44:4 1998.12[民87.12] |
頁 次 | 頁62-75 |
分類號 | 443 |
關鍵詞 | 移流; 延散; 一維; 分離運算子; Advection; Dispersion; One-dimensional; Split operator; Holly-preissmann; Characteristics; Crank-nicolson; |
語 文 | 中文(Chinese) |
中文摘要 | 以有限差分數值方法模擬污染物的傳輸現象,存在著模擬結果有嚴重的數值延散 誤差,尤其是當濃度梯度大的情況。這個問題雖已被前人克服,然而通常需要較複雜的數值 方法(Li, et al, 1992)。本文以分離運算子觀念(Split Operator)將污染物的傳輸視為在同一個時 段中,移流與延散過程乃先後發生,如此可將傳輸方程式拆為移流過程與延散過程而分別以 基本的數值方法求解,在適當的參數條件下仍可令數值結果符合物理過程。本文以單網路 Holly-Preissmann特性線法來模擬污染物的移流傳輸過程,以Crank-Nicolson有限差分法模 擬延散傳輸過程。為能掌握數值模式的準確性與參數範圍,本文以標準試例中高斯分佈與階 梯分佈兩種初始濃度型態來測試,改變主要參數的大小來估測模式的準確性。所得的結果, 顯示模擬的瓶頸乃在移流過程,因此在可蘭數數值為1時,模式模擬結果可達最高的精度。 在可蘭數小於1以空間軸差分之模擬結果優於可蘭數大於1以時間軸差分之模擬結果。物理 上延散效應的存在,反而使得數值模擬較容易,可蘭數的適用範圍也可因而擴大。然而若以 相同的數值Peclet數(移流/延散 比例)而言,模式解析度(網格數)的增加又要比延散效應的存 在更能改善精度。 |
英文摘要 | Originated from split-operator concept, the one-dimensional advection/ dispersion equation describing distribution of concentration is solved in two steps. In the first step, only advection process is considered. The second step is within the same time interval, in which dispersion operator is performed on the advected resulting distribution. Holly-Preissmann scheme, a method of characteristic line, is adopted in the advection step. Crank-Nicolson scheme, a finite difference method, is used in the dispersion step. Benchmark problems with initial distributions of continuous Gaussian as well as discontinuous step shapes are tested. Accepted ranges of numerical parameters such as Crount numbers, Peclet numbers, and grid sizes are investigated in different circumstances. Since advection is found to be the bottle-neck process, Courant number is best to be kept as near 1.0 as possible. Cases with Courant number less than 1.0 performs better than that larger than 1.0, which implies that time-line interpolation is less accurate. Existing of physical dispersion (lower the Peclet numbers)actually improves computation effort and, therefore, increases the acceptable range of numerical parameters. Refinement of grid is found to be even effective than lower the Peclet numbers. |
本系統中英文摘要資訊取自各篇刊載內容。