量子統計稀薄氣體直接解法研究

Abstract

本文對於求解不同統計粒子流動問題提出一種新的通用計算解法。三種統計粒子 分別遵循Maxwell-Boltzmann、 Fermi-Dirac 以及 Bose-Einstein 統計。基於BGK 方法 (Bhatnagar-Gross-Krook method) 利用鬆弛時間近似處理相空間中應用於廣 泛Knudsen 數範圍內的氣體動力學問題。 本數值方法使用離散座標法(discrete ordinate method)對於半古典分布函數中的速 度空間做處理，使其轉換為一組含有源項的純量守恆律。使用高階方法包含全變 量消逝法(Total Variation Diminishing / TVD) 、基本不振盪法 (Essentially Non-Oscillatory)、加權型基本不振盪法 (Weighted Essentially Non-Oscillatory schemes) 以及CE/SE (Conservation Element/Solution Element) 方法來演算物理 空間與時間內的解。本文發展出在卡氏座標以及通用座標中求解多維度問題的顯 式與隱式算則。 數值實驗包含 (1) 不同統計粒子在各種Knudsen 數和鬆弛時間下的一維震波管 問題 (2) 正交和非正交格點下使用不同鬆弛時間的二維穩態及暫態氣體流動問 題。針對初始及邊界值問題測試了各種顯式及隱式求解器。由數值實驗結果可知， 本文所提出的數值方法為可行且可靠的方法。

This dissertation aims to provide a novel unified computational method to solve gas flows problems of arbitrary particle statistics namely, Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics. The relaxation time approximation based on Bhatnagar-Gross-Krook method was implemented in phase space in order to tackle gas dynamics problems in a wide range of Knudsen numbers. The numerical method is based on the discrete ordinate method to render the velocity space of the semiclassical distribution function resulting in a set of scalar conservation laws with source terms. High resolution methods comprising Total Variation Diminishing (TVD), Essentially Non-Oscillatory, Weighted Essentially Non-Oscillatory schemes (ENO/WENO) and Conservation Element / Solution Element (CE/SE) were used for evolving the solution in physical space and time. The classes of explicit and implicit schemes for solving multi-dimensional problems in Cartesian and general coordinate were developed. The computational experiments include (1) One-dimensional shock tube problem in which the range of Knudsen numbers and relaxation times were tested for gases obeying the three statistics, (2) Two-dimensional gas flow problems covering both transient and steady state cases on Cartesian and curvilinear grids and various relaxation times. The variants of explicit and implicit solvers were tested on initial and boundary value problems. The results from the computational experiments shows the feasibility and robustness of the current method.