伸縮迭代法於基爾霍夫型問題上之應用及泊松-波茲曼型問題的數值模擬

Abstract

這篇論文包含兩個主題：伸縮迭代法 (Scaling Iterative Algorithm, SIA) 於基爾霍夫型問題 (Kirchhoff-type problems) 之應用，以及泊松-波茲曼型問題 (Poisson-Boltzmann-type models) 的數值模擬。基爾霍夫型問題是一個多解問題，其除了擁有零解之外，在不同條件下另會有不同的非零解；如球對稱解。若使用一般的數值方法求解基爾霍夫型問題，則總是會得到零解；其非零解的高度不穩定性，即使選為初值代入求解器 (solver) 迭代，都有可能收斂至零解或發散。伸縮迭代法很好地解決了此類問題。我們先使用伸縮迭代法研究萊恩-埃姆登問題 (Lane-Endem problem)，在二維環區域上，求出數個多峰正解。而在基爾霍夫類問題上，由於其非線性及非局部性，我們則發展了隱式伸縮迭代法 (Implicit-SIA) 來求其正解。 在泊松-波茲曼類問題上，我們首先從數值上檢驗幾個離子通道模型的差異。我們從數值上驗證了 PB_ns 模型在某些條件下可化約為李波模型 (Li Bo’s model)。其次，對於具空間效應的穩態 Poisson-Nernst-Planck 模型，我們在給出其代數方程部分的多解之充要條件並確認多解之位置後，實際計算出了其數值多解。在這兩個案例中，對其模型內代數方程組的求解方法亦有進一步的討論。 本論文中關於一維微分方程計算使用了 MATLAB，而高維情況則使用流體力學軟體 OpenFOAM。計算結果以 ParaView 製圖呈現。

In this thesis, we first revisit multi-peak solutions to the Lane-Endem problem as in [7]. In order to find numerical solutions, the Scaling Iterative Algorithm (SIA) is implemented in our solvers. By adding different symmetrical restrictions on the numerical solutions, more multi-peak solutions to the Lane-Endem problem are found. After the Lane-Endem problem, we investigate into nontrivial solutions of the Kirchhoff-type problems. The SIA is again implemented in the solver for positive solutions which standard iteration methods failed to compute. Implicit SIA (ISIA), an improvement of the SIA, is developed due to the nonlinear and nonlocal nature of the Kirchhoff -type problem. Three implementations of the ISIA solvers are also discussed. The second part of this thesis is on numerical approaches to Poisson-Boltzmann-type equations. We develop solvers for Poisson-Boltzmann-type equations and give numerical supports to a few theoretical results. We check the fact that the PB_ns model (4.19) can be reduced to Li Bo's model (4.20) (also see [12][13]). Such solvers are also used to study multiple solutions of steady-state Poisson-Nernst-Planck equations with steric effects (PNP-steric model), whose existence was proven in [3]. Two distinct solutions of the PNP-steric model are numerically found and plotted after we clarity where they might appear. The efficiency and complexity of the solvers are also discussed. Most problems mentioned above are modeled and solved in OpenFOAM, which is a free, open source software for computational fluid dynamics problems. Others are solved in MATLAB. Profiles of all the solutions are delicately plotted.