加速收斂法用於計算三種地下水解析解上

Abstract

有些地下水問題的解析解，包含無限級數及積分式。若積分式中被積函數為震盪函數，則積分式通常可轉換成無限級數。有些無限級數因收斂緩慢，要估算很多項，而有耗費計算時間及/或估計值經度不高等問題。本文主要目的，在探討並比較三種加速收斂法：Euler 轉換法、Aitken轉換法、及Shanks轉換法，以加速估算得解析解的數值。在取用相同之收斂準則下，分別使用此三轉轉換方法，首先應用於兩種無限級數，接著再計算定水頭邊界條件下抽水洩降解析解之數值[Carslaw and Jaeger, 1959]，也計算自由含水層之洩降量解析解[Neuman, 1974]，最後應用於徑向流解析解上[Hsieh, 1986]；結果顯示Shanks及Aitken 兩種轉換法，加速收斂的效果較佳。

Some analytical solutions in the field of groundwater involve an infinite series or an integral. If the integrand is an oscillatory function, then this integral may be transformed to an infinite series. However, the infinite series may sometimes have the problem of slow convergence. The evaluations of the integration may therefore be not straightforward and time-consuming. Under the selected convergence criterion, this thesis initially employs three transform methods including the Euler transform, the Aitken’s transform, and Shanks’ transform to accelerate the evaluation of two different types of infinite series. These methods are also applied to evaluate the solution for groundwater flow under a constant-head boundary condition [Carslaw and Jaeger, 1959], the analytical solution of the aquifer drawdown for the unconfined flow equation [Neuman, 1974], and the analytical solution of radial dispersion problem in groundwater [Hsieh, 1986]. The results indicate that both the Aitken’s transform and Shanks’ transform are efficient in acceleration the evaluation while applied for those three drawdown solutions.