• 作者： Chen, Ming-Quayer; Hwang, Chyi; Shih, Yen-Ping
• 中文摘要： Stefan問題為一非線性暫態問題,其空間領域的邊界會隨著時間移動。移動邊界不僅妨礙了解析解的獲得,並且導致求數值解的過程中,必須以疊代來計算邊界移動的時間步幅。Gupta及Kumar所提出的座標變換法可使邊界移動大小與空間格點尺度大小無關。故在求一維Stefan問題時,藉由此法可免去疊代計算的問題。本文應用Galerkin方法結合座標變換法將一維Stefan問題轉變成起始值問題,並利用Daubechies所提出的正交小波函數,作為Galerkin方法的基底函數。為了正確計算Galerkin公式化所需的積分值,我們推導了以小波函數為基底之相關的微分與積分計算公式。此外,為了避免使用有限差分法在準確度控制上的困難,本文採用了可依準確度而自動調整積分步幅大小的數值法來解起始值問題。本文也利用所提出的方法來解以冰-水為媒介的熱傳問題,並與有限差分法的數值解比較,結果顯示本法具有較佳的準確性。
• 英文摘要： Stefan problems are nonlinear transient problems which involve a domain whose boundary moves in time. The presence of moving boundary not only hinders the analytical solution to Stefan problems but also leads a numerical procedure to involve an iterative computation of the time- step for a given advancement of the moving boundary. The coordinate transformation approach proposed by Gupta and Kumar (1980) dissociates the boundary advancement from the size of space mesh and thus eliminates the iteration steps for the numerical solution of a 1-D Stefan problem. In this paper, a Galerkin method along with the coordinate transformation is applied to convert 1-D Stefan problems into initial-value problems. The class of compactly supported orthonormal wavelets developed by Daubechies( 1988) will be adopted as the Galerkin bases in the spatial domain. For an exact Galerkin formulation, computational algorithms for exactly evaluating the integrals of wavelet bases and their derivatives are derived. In order to avoid the difficulty of accuracy control associated with the finite-difference methods, it is suggested to solve the resulting initial-value problems by the numerical integration scheme that has accuracy control by automatically adjusting the time step. The method is illustrated with solving the Stefan problem concerning the heat transfer in an ice-water medium. The obtained numerical results are more accurate than those obtained by the finite- difference methods.
• 中文關鍵字： 小波; Stefan問題; 非線性問題; 暫態; 座標變換
• 英文關鍵字： Wavelet; Stefan Problem; Nonlinear Problem; Transient(Transition State); Coordinate Transformation