移動邊界截斷格點法是一種能自適決定計算區域邊界，並顯著降低格點使用率的固定網格有限差分算則。隨著格點截斷和邊界外插，相空間格點被動態啟用、關閉，以高效並準確積分動力學方程式。在本研究中，截斷格點法被拓展到求解二維相空間中的波茲曼–BGK方程式，並應用於三個領域⸺化學物理、計算流體力學以及半導體物理。在化學物理案例上，吾人首先檢驗截斷格點法，能在斯莫魯霍夫斯基極限下重現擴散動力學，印證其滿足擴散漸進解；隨後，計算欠阻尼條件下、雙阱位能中的逃逸問題，截斷格點法將系統演化至波茲曼分布，並給出和全格點法一致的穿透機率動力學。在計算流體力學案例上，截斷格點法成功描述局域平衡起始態的弛豫過程，與文獻中援引高階傅立葉擬譜法所建構之基準解諳合，並將系統演化至正確的全域平衡態；此外，截斷格點法被進一步應用於更複雜的遠平衡起始態之混合體系問題，計算出與階層式高階不連續伽遼金隱式－顯式算法相吻合的流體變數模式，並能得出流體力學變數均勻分布之平衡態解。在半導體物理案例上，透過半導體波茲曼–泊松系統，以截斷格點法研究砷化鎵二極體，對於常數弛豫時間模型，截斷格點法精確描述準彈道電子之輸送現象，並給出與五階加權本質無振盪算則一致的穩態解；吾人隨之採用更加真實的極性光學聲子模型，得出與常數弛豫時間模型有定量差異之結果，而截斷格點法之通道模擬結果再次與全格點法相吻合，且其精度足以區分出不同模型之區別。在所有數值案例中，截斷格點法皆成功透過減少格點的使用，降低數值計算之實時成本，展現截斷比例越高則省時益多，截斷比例越低則準度越精之趨勢；綜合考量，移動邊界截斷格點法可作為求解波茲曼–BGK方程式之高效能全動力學算則。

The moving boundary truncated grid (TG) method is a fixed-mesh finite-difference scheme that self-adaptively determines the boundary of the computational domain, leading to a substantial reduction in grid-point usage. By employing grid truncation and boundary extrapolation, grid points are dynamically activated and deactivated, allowing for the integration of dynamical equations with remarkable accuracy and high efficiency. In this study, the TG method is extended to solve the Boltzmann–BGK equation in a two-dimensional phase space, and we explore its applications in chemical physics, computational fluid dynamics (CFD), and semiconductor physics. In chemical physics, we validate the TG method by reproducing diffusive dynamics in the Smoluchowski limit. The method is confirmed to satisfy the asymptotic solution derived from the diffusion equation. Furthermore, the TG method is applied to investigate the Kramers escape problem in the underdamping regime. It evolves the system toward the Boltzmann distribution as an equilibrium solution. The transmission probability obtained from the TG method aligns well with that provided by the conventional full-grid (FG) method. In CFD, the TG method successfully depicts the relaxation process of a local-equilibrium initial state, in accordance with the benchmark solution constructed from a high-order Fourier pseudo-spectral method in the literature. It reliably evolves the system toward the expected global-equilibrium state. The method is further applied to a more complex mixed-regime problem with a far-from-equilibrium initial state. It captures the pattern of hydrodynamic variables comparable to that obtained from a hierarchical high-order discontinuous Galerkin implicit-explicit scheme. The method successfully predicts a global-equilibrium solution characterized by spatial homogeneity across all hydrodynamic variables. In semiconductor physics, the TG method is utilized to study a gallium-arsenide diode, modelled by a Boltzmann-Poisson system. For the constant relaxation-time models, it accurately describes the transport phenomena of quasi-ballistic electrons and provides a steady-state solution consistent with that furnished by the fifth-order weighted essentially non-oscillatory scheme. Nevertheless, we adopt a more realistic polar-optical-phonon model, which yields results with quantitative differences from that of the constant relaxation-time models. The TG solver demonstrates excellent agreement with the FG solver in channel simulations, effectively distinguishing between the two models. In all numerical examples, the TG method manages to reduce the computational costs. The TG parameters directly influence the trade-off between accuracy and efficiency. Heavier truncation results in greater time savings, whereas lighter truncation produces more accurate results. In summary, the TG method serves as an efficient full-kinetic solver for the Boltzmann–BGK equation.

第一章 理論基礎 1
1-1. 理論力學概論 1
1-1-1. 哈密頓力學 1
1-1-2. 劉維爾定理 3
1-2. 氣體動力理論 6
1-2-1. BBGKY層系 6
1-2-2. 物理圖像及分子混沌假設 8
1-2-3. 波茲曼輸送方程式 11
1-3. 波茲曼方程式之性質和模型方程式 13
1-3-1. 碰撞不變量、動差方程式和守恆律 13
1-3-2. 馬克士威分布和熱平衡 18
1-3-3. 線性化碰撞算符之數學性質 21
1-3-4. 弛豫時間模型與BGK方程式 25
第二章 研究方法 27
2-1. 數值微分方程 27
2-1-1. 數值微分：有限差分近似 27
2-1-2. 數值積分：牛頓-寇次公式 32
2-1-3. 起始值問題：龍格-庫塔法 37
2-1-4. 起始‐邊界值問題：有限差分算則 41
2-2. 移動邊界截斷格點法 46
2-2-1. 計算流程 47
2-2-2. 對於相空間分佈函數時間演化之應用 49
第三章 研究成果 60
3-1. 化學物理學上的應用 60
3-1-1. 斯莫魯霍夫斯基極限 60
3-1-2. 克拉默斯之逃逸問題 100
3-2. 計算流體力學上的應用 114
3-2-1. 局域平衡態之弛豫動力學 115
3-2-2. 遠平衡態之混合體系問題 125
3-3. 半導體物理學上的應用 135
3-3-1. 常數散射速率模型 147
3-3-2. 極性光學聲子模型 159
第四章 結論與未來研究方向 165
參考文獻 167

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