雙曲線型線性偏微分方程式界面問題的數值方法

Abstract

此篇論文主要是在探討線性雙曲線型偏微分方程式系統界面問題的數值方法。在界面上，通常需要滿足邊界條件，而系統的參數在此也是不連續的。這樣的問題在物理模擬中很常見。譬如說，電磁波從一個介質傳遞至另一個介質中。雙曲線型偏微分方程的標準數值方法通常只試用於係數為連續函數的情況，而在界面問題上，這些方法會失效。因此，在界面週邊必須要做特殊處理。我們介紹兩種以卡式坐標系網格為基準的方法 - 'Ghost Fluid'方法及 'Immersed Interface'方法 - 來處理不連續的邊界條件。

In this thesis, we investigate the numerical techniques for solving interface problem of linear hyperbolic system of equations with piecewise constant coefficients and jump conditions across the interface. Such problem arises naturally in practical physics, for example, electromagnetic waves propagating from one material to the other. Standard numerical techniques for solving hyperbolic systems fail near the interface, and special treatments must be offered. Two Cartesian-based methods, 'ghost fluid method' and 'immersed interface method', are introduced to catch the jump discontinuity.