利用耦合界面法解決多原子問題

Abstract

本篇論文是關於研究泊松-玻爾茲曼方程式(PBE) 的數值解法。在生物物理學當中，PBE 是用來以描述分子在水溶液中的靜電勢能。在玻爾茲曼方程式的數值計算上，我們主要會碰到的兩個困難點：第一點是電荷奇異性，第二點是表面奇異性。前者主要是來自方程當中用來表示分子中電荷的delta 函數(單位脈衝函數)；後者來自複雜的分子表面，介電係數高度的落差。 關於第一個困難點，我們引進一個點電荷在真空中的位能函數來解決，如在[21] 所提議的。第二點困難，我們提出耦合界面法(CIM) 來對付。對於處理橢圓界面問題，它是一種簡單健全的方法[1]。 數值測試顯示出耦合界面法比其他界面問題的解法來的優異。它在位能和梯度上都可以達到二階收斂。

In this master thesis, we study Poisson-Boltzmann equation (PBE) numerically.In biophysics, the PBE is used to describe the electrostatic potential for molecules in solvent. Two difficulties encountered as we solve PBE numerically: the charge singularities and the surface singularities. The former comes from the point charges of molecule, they are the delta functions in PBEs. The latter comes from the complicated molecular surface, across which the dielectric coefficient has jump. The 1st difficulty is resolved by introducing a potential induced by those point charges in vacuum, as proposed in [21]. The 2nd difficulty is resolved by using the coupling interface method (CIM) [1], which is a simple and robust method for solving elliptic interface problems. Numerical tests show that the performance of CIM is superior to other interface methods. It is second for both potential and its gradient.