高度不均勻介質中的橢圓形方程與拋物線方程的數值誤差分析

Abstract

這是一個二年期的計劃。我們希望探討計算非均勻橢圓 方程式與非均勻抛物線方程式的方法及這計算方法的誤差 估計問題。以污染源(如化學工廠排放的廢水，核電廠的廢 棄物等) 在地底下的擴散為例。地底下的縫隙結構與地質性 質在不同的位置就有很大的差異，並且污染源擴散時在污染 源的附近或較遠處會有平流與對流現象發生。因此描述此現 象的數學模式就包含了非均勻橢圓方程式與非均勻抛物線 方程式。若是能發展出有效率的計算方法，則可借用計算機 的計算功能幫助我們更清楚了解多孔介質中的多相流在微 觀模式下的細微變化。 之前的計劃大多討論描述多相流的微觀模式與其宏觀 模式之間的關係，也了解一些多相流的微觀模式的均勻收斂 的結果。若是能夠有效率的計算多相流的微觀模式的解。對 多相流在多孔介質中的運動一定能有更進一步的了解。

This is a two-year project. We plan to find efficient numerical schemes to compute the solutions of non-uniform elliptic equations and non-uniform parabolic equations. The contaminant transportation and the multi-phase flows in highly heterogeneous media are strongly related to the geology of soil and are complicated. Flows in the media may show many different time-scale phenomenon. Their corresponding microscopic models usually consist of convection and diffusion equations. In other words, their mathematical models contain non-uniform elliptic equations and non-uniform parabolic equations. Therefore, this research helps understanding waste contaminant transport in soil and flows in porous media. In previous projects, we studied models for multi-phase flow problems in microscopic level and in macroscopic level. We also derive uniform estimates for solutions of non-uniform elliptic equations. If it is possible to develop highly efficient numerical schemes to compute the solutions of non-uniform elliptic equations and non-uniform parabolic equations, we then have powerful tools to tackle multi-phase flow problem in porous media.