宇宙射線(cosmic rays, CRs)在星系及星系團的形成與演化扮演著重要的角色，他們可以透過在天體物力的非碰撞震波(collisionless shock)中的擴散衝擊加速機制(diffusive shock acceleration, DSA)來加速粒子。除此之外，透過物理特性在震波中的跳躍條件來實施震波尋找演算法並且評估此方法的效益是非常重要也具有挑戰性的。因此，我們的研究主要是在具有宇宙射線模組的FLASH代碼中，並在子網格模型上注入宇宙射線的能量，此能量多寡則是透過擴散加速機制的傳遞來決定。我們使用二維Sedov爆炸以及Sod震波管測試問題來個別產生由能量所驅動的球形震波以及平面震波。在我們的研究中可以透過震波尋找演算法有效的找到震波位置及震波表面、以及透過溫度特型的跳躍條件來估算馬赫數，並且相較於壓力特性的跳躍條件較為準確。而此代碼可以廣泛運用在各項包含模擬宇宙射線的系統中，像是超新星爆炸遺跡、電波星系以及銀河風等。

Cosmic rays (CRs), which are relativistic charged particles accelerated via diffusive shock acceleration (DSA) within collisionless shocks, play an important role in formation and evolution of galaxies and clusters. For modeling shock-accelerated CRs in hydrodynamic simulations, it is crucial and challenging to implement a shock-finding algorithm based on jump conditions across the shocks and evaluate their performance. Therefore, our aim of this study is to implement a subgrid model for CR injection due to shock acceleration in the CR module in the FLASH code and estimate how much dissipated energy can be transferred to CRs through DSA. In addition, we use 2D Sedov-explosion and Sod shock-tube test problems to verify our algorithms and to study the influence of CRs in energy-driven spherical shocks and panel shocks, respectively. In our simulations, we verified that our shock-finding algorithm can effectively find locations of the shock zone and shock surface. Criterion based on temperature jumps predicts more accurate Mach numbers than pressure jumps. This code will have a broad application to the modeling of CRs in various systems, including supernova remnants, radio galaxies, and galactic winds.

Abstract（Chinese）----------------------------------------------- I
Acknowledgements（Chinese）------------------------------------ II
Abstract ----------------------------------------------------------- III
Acknowledgements ------------------------------------------------ IV
Contents ---------------------------------------------------------- V
List of Figures ----------------------------------------------------- VII
List of Tables ------------------------------------------------------ IX
List of Algorithms -------------------------------------------------- X
1. Introduction-------------------------------------------------------1
1.1 Introduction to cosmic rays---------------------------------------1
1.2 Collisionless shocks and particles acceleration mechanisms -----2
1.3 Recent simulations about shock finding algorithm and implementation of shock-accelerated CRs-----------------------------------------------------------------4
1.4 Goals and motivations -------------------------------------------5
2. Simulations and Methods -----------------------------------------9
2.1 The FLASH code ------------------------------------------------9
2.2 Shock-finding algorithm ---------------------------------------10
2.2.1 The Rankine–Hugoniot jump conditions -----------------------11
2.2.2 Shock zone regions -------------------------------------------12
2.2.3 Shock surface region------------------------------------------15
2.2.4 Mach number calculation -------------------------------------15
2.3 Diffusive shock acceleration of cosmic rays. --------------------16
2.4 Setup of the test simulations ------------------------------------18
2.4.1 Numerics for 2D Sedov explosion ------------------------------19
2.4.2 Numerics for 2D Sod shock-tube ------------------------------20
3. Results ----------------------------------------------------------23
3.1 Sedov explosion ------------------------------------------------23
3.2 Sod shock-tube -----------------------------------------------28
4. Discussion -------------------------------------------------------32
4.1 Resolution study for the Sedov test ------------------------------32
5. Summary --------------------------------------------------------37
Bibliography --------------------------------------------------------39

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