原文信息 出处：Numerical Simulations in Cosmology: Chapter 3 — Hydrodynamic methods and sub-resolution models for cosmological simulations 作者：Milena Valentini (Universitá degli Studi di Trieste / INAF), Klaus Dolag (LMU München / MPI for Astrophysics) arXiv：arXiv:2502.06954 页数：61 页，14 幅图 翻译说明： 中文翻译，段落对照，上面是中文，下面是英文。Chinese translation, paragraph by paragraph, with Chinese on top and English below. 虚线下划线 为个人添加的注释或评论。文献引用如 [1] 可点击复制完整引用信息。 ...
引力是驱动结构形成的基本力，因此大多数宇宙学模拟的核心构建模块便是 $N$ 体程序（关于替代方法的讨论，参见[1] ）。$N$ 体程序的目标是研究自引力、无碰撞系统的非线性动力学演化（综述见[2] ,[3] ）。Being gravity the force that drives structure formation, the building block of the majority of cosmological simulations is an N-body code (see e.g. [1] for a discussion on alternative approaches). The goal of an N-body code is to investigate the non-linear dynamical evolution of a self-gravitating, collisionless system (see [2] ,[3] , for reviews). ...
宇宙中引力成团具有高度非线性，这使数值模拟成为详细追踪星系和星系团等形成结构之演化的唯一手段。然而，这需要捕获巨大的空间和时间动力学范围。例如，结构等级式并合跨越的物理尺度从星系内部的亚 kpc 尺度，一直延伸到数百兆秒差距——后者是宇宙中最大的相干尺度。在这方面，基于 $N$ 体和流体动力学技术的现代宇宙学模拟程序，最适合在结构等级式形成过程中精确追踪暗物质和气体在其全部复杂性下的联合动力学。As a result of the high non-linearity of the gravitational clustering in the Universe, numerical simulations are the only method to follow the evolution of the forming structures like galaxies and galaxy clusters in detail. However, a huge dynamic range in space and time has to be captured. For instance, the range of physical scales over which the hierarchical assembly of structures develops spans from sub-kpc scales in galaxies up to several hundreds of megaparsecs, the latter being the largest coherent scale in the Universe. Here, modern cosmological simulation codes based on N-body and hydrodynamics techniques are best suited to accurately follow the joint dynamics of DM and gas in their full complexity, during the hierarchical build-up of these structures. ...
过去几十年中，人们发展了多种数值方案来求解描述宇宙重子成分与无碰撞暗物质的耦合方程组。绝大多数重子（即气体）可被描述为理想流体，其演化由一组方程——即欧拉方程——所支配。积分上述方程的流体求解器可归为两大类，如Fig. 2所示：粒子方法（离散化质量，见[47] 及其参考文献）和网格方法（离散化计算域，见[48] 及其参考文献）。近年来又涌现出新的求解器：它们融合了两种方法的特征（见[49] 及其参考文献），将在后续章节中详细讨论。A variety of numerical schemes has been developed in the past decades to solve the coupled system of equations describing the baryonic content of the Universe and the collisionless DM. The majority of the baryons (i.e. gas) can be described as an ideal fluid, whose evolution is ruled by a set of equations, namely, the Euler equations. The hydro solvers which integrate the aforementioned equations fall into two main categories, that are summarized in Fig. 2: particle methods, which discretize mass (see [47] and references therein), and grid-based methods, which discretize the computational domain (see [48] and references therein). Recently, additional solvers have been developed: they combine characteristics of both methods (see [49] and references therein) and will be discussed in detail in the following sections. ...
Eulerian (grid) methods 网格方法在结构或非结构化网格上求解欧拉方程（参见Fig. 2），用以表示流体。其中，原始变量描述流体的热力学性质（如 $\rho$、${\vec v}$ 或 $P$），而守恒变量则定义守恒定律（如 $\rho$、$\rho{\vec v}$ 或 $\rho u$）。早期研究者尝试使用中心差分格式：流体仅以网格中心的数值表征，导数则通过有限差分表示求得（参见例如 [51] ）。这些方法采用人工黏性来处理激波（类似于§3.6中描述的平滑粒子流体动力学方法），否则在出现间断的区域会失效。此外，从构造上讲，它们仅具有一阶精度。Grid-based methods solve the Euler equations (see Fig. 2) based on structured or unstructured grids, representing the fluid. While primitive variables encode the thermodynamic properties of the fluid (e.g., $\rho$, ${\vec v}$, or $P$), conservative variables define the conservation laws (e.g., $\rho$, $\rho{\vec v}$, or $\rho u$). Early attempts were made using a central difference scheme, where fluid is only represented by the centered cell values and derivatives are obtained by the finite-difference representation (see, for example, [51] ). These methods use artificial viscosity to handle schocks (similar to the smoothed particle hydrodynamics method described in §3.6), as they would otherwise break down in regimes where discontinuities appear. Also, by construction, they are only first-order accurate. ...
Adaptive mesh refinement 为拓宽数值格式的动力学范围，多种网格程序已采用网格细化策略（例如 ART、RAMSES、ENZO 和 FLASH）。多数情况下采用简单的密度（即每个网格的质量）判据：若某网格内的质量超过阈值 $m \equiv \rho , \Delta x^3 > m_{min}$， 则将该网格分割为多个（例如八个）子网格，并将内部属性从原网格插值到新子网格上。这确保了引力质量（即引力源）在计算域内均匀分布。如此一来，底层网格便以准拉格朗日方式跟随质量流演化，如 Fig. 4 左图所示，该图展示了宇宙学模拟中细化网格的典型结构。To widen the dynamical range of the numerical schemes, mesh refinement strategies have been applied in several grid codes (e.g., ART, RAMSES, ENZO, and FLASH). In most of the cases, a simple density (e.g., mass per cell) criterion is used. If the mass within one cell exceeds a certain threshold, $m \equiv \rho \, \Delta x^3 > m_{min}$, the cell is divided in multiple (e.g., eight) sub-cells and the internal properties are interpolated from the original cell onto the new sub-cells. This ensures that the gravitational mass (e.g., the source of gravity) is homogeneously distributed within the computational domain. In this way, the underlying grid evolves in a quasi-Lagrangian fashion following the mass flow, as illustrated in the left panel of Fig. 4, which shows the typical structure of the refinement grid in a cosmological simulation. ...
Lagrangian methods and Smoothed Particle Hydrodynamics 经典的拉格朗日方法是所谓的光滑粒子流体动力学方法（SPH；[78] ,[79] ），它求解Euler方程的拉格朗日形式（参见Fig. 2），并可在高密度区域获得良好的空间分辨率。The classical Lagrangian method is the so-called Smoothed Particle Hydrodynamics method (SPH; [78] ,[79] ), which solves the Lagrangian form of the Euler equations (see Fig. 2) and can achieve good spatial resolution in high-density regions. ...
Moving mesh methods 研究者投入了大量精力，将§3.4中描述的Euler方法重新表述为Lagrange网格方法。移动网格流体动力学在宇宙学中应用的更多细节，可参见[101] 的开创性工作及其参考文献。这种早期方法从规则网格出发，然后通过跟随流体流动使网格变形。Euler方程通过计算网格边界上的通量来演化。宇宙学模拟中得到的网格示例见Fig. 6左图。该技术在实际应用中的一个缺点（或挑战）是单个网格单元可能严重变形和拉伸。现代方案通过基于Voronoi或Delaunay镶嵌构建非结构网格，规避了这一问题（参见[102] 及其参考文献）。基于网格生成点$\vec{r}_i$和$\vec{r}_j$的网格单元相关几何结构如Fig. 6右图所示。然后必须在界面的质心处计算通量（注意，该质心不一定位于两个网格生成点之间的直线上，如虚线所示） $$ \mathbf{Q}i^{(n+1)} = \mathbf{Q}i^{(n)} - \Delta t \sum_j A{ij}\hat{\mathbf{F}}{ij}^{(n+1/2)}. $$ 该界面的运动$\vec{w}$由速度$\vec{w}_i$和$\vec{w}_j$唯一定义，并且必须在旋转坐标系($x’,y’$)中使用Riemann求解器计算通量。该技术的详细描述及其在测试问题中的表现，可参见[102] 。Substantial effort has gone into reformulating Eulerian methods as described in §3.4 into Lagrangian mesh approaches. More details on the idea of hydrodynamics on moving mesh for cosmological application can be found in the pioneering work by [101] and references therein. This early approach started from a regular mesh which then, by following the flow of the fluid, was deformed. The Euler equations were evolved by calculating the fluxes across the cell borders. An example of the resulting mesh for a cosmological simulation can be seen in the left part of Fig. 6. One disadvantage (or challenge) of this technique in practical applications is that individual cells can be extensively deformed and stretched. Modern schemes circumvent this problem by constructing an unstructured mesh based on a Voronoi or Delaunay tessellation (see [102] and references therein). The relevant geometry of the cells, based on the mesh generating points $\vec{r}_i$ and $\vec{r}_j$ is illustrated in the right part of Fig. 6. The fluxes then have to be calculated at the centroid of the interface (note that this is not necessarily on the straight line between the two mesh generating points, as indicated by the dotted line) $$ \mathbf{Q}_i^{(n+1)} = \mathbf{Q}_i^{(n)} - \Delta t \sum_j A_{ij}\hat{\mathbf{F}}_{ij}^{(n+1/2)}. $$ The motion $\vec{w}$ of this interface is uniquely defined by the velocities $\vec{w}_i$ and $\vec{w}_j$, and the fluxes have to be calculated with the Riemann solver in the rotated frame ($x',y'$). A detailed description of this technique along with its performance in test problems can be found in [102] . ...
Meshless methods 最近，一类新的拉格朗日方法——即所谓的无网格公式——在天体物理领域得到了发展。更多细节见[102] 、[104] 、[105] ，这些工作继承了[106] 、[107] 、[108] 更早的开创性研究。简而言之，推导从积分形式 $$ \int[u(\vec{x},t)\dot\phi(\vec{x},t) + \vec{F}(u,\vec{x},t)\cdot\nabla\phi(\vec{x},t) + S(\vec{x},t)\phi(\vec{x},t)],d\vec{x},dt = 0 $$ 出发，它是标量守恒律 $$ \frac{\partial u}{\partial t} + \nabla\cdot(\vec{F} + \vec{a}u) = S $$ 的积分形式。其中，$u(\vec{x},t)$是一个标量场，$S(\vec{x}, t)$是其源项，$\vec{F}(u,\vec{x},t)$是在以速度$\vec{a}(\vec{x}, t)$运动的参考系中的通量，而$\phi(\vec{x},t)$是空间和时间上的任意可微函数，由此定义了随体导数$\dot\phi(\vec{x}, t) = \partial\phi(\vec{x}, t)/\partial t + \vec{a}(x,t)\cdot\nabla\phi(\vec{x}, t)$。A new class of Lagrangian methods, the so-called meshless formulations, have been recently developed for astrophysical problems. More details can be found in [102] ,[104] ,[105] , which follow earlier, pioneering work by [106] ,[107] ,[108] . In short, the derivation starts from the integral form $$ \int[u(\vec{x},t)\dot\phi(\vec{x},t) + \vec{F}(u,\vec{x},t)\cdot\nabla\phi(\vec{x},t) + S(\vec{x},t)\phi(\vec{x},t)]\,d\vec{x}\,dt = 0 $$ of a scalar conservation law $$ \frac{\partial u}{\partial t} + \nabla\cdot(\vec{F} + \vec{a}u) = S \,. $$ Here, $u(\vec{x},t)$ is a scalar field, $S(\vec{x}, t)$ is its source, $\vec{F}(u,\vec{x},t)$ is its flux in a frame moving with velocity $\vec{a}(\vec{x}, t)$, and $\phi(\vec{x},t)$ is an arbitrary differentiable function in space and time leading to the advective derivative $\dot\phi(\vec{x}, t) = \partial\phi(\vec{x}, t)/\partial t + \vec{a}(x,t)\cdot\nabla\phi(\vec{x}, t)$. ...
Code comparison in galaxy and cluster simulations 前文所述的欧拉和拉格朗日方法，在理论上应用于同一问题时应给出相同的结果。为验证代码能否正确积分流体力学方程组，通常会用已知解析解的问题来测试。常见的测试问题包括激波管或球对称坍缩问题。The Eulerian and Lagrangian approaches described in the previous sections are theoretically supposed to provide the same results when applied to the same problem. To verify that codes succeed at correctly integrating the set of hydrodynamical equations, they are usually tested against problems whose solution is known analytically. In practice, these test problems are shock tubes or spherical collapse problems. 不过，理想化的流体动力学测试——例如多相流体的相互作用 [109] ——通常会揭示不同方法所得结果之间的根本差异。造成这些差异的原因可能是基本方程的表述形式（例如经典SPH中不存在混合）、离散化方式（例如SPH表述中的体积偏差），或是数值误差的影响（例如网格代码中因重构误差而导致的平移不变性偏离）。However, idealized hydrodynamical tests like the interaction of multi-phase fluids [109] often reveal fundamental differences among results obtained with different methods. Such differences can be driven by the formulation of the underlying fundamental equations (like no mixing in classical SPH), by the discretization (like the volume bias in SPH formulations) or they can be due to the influence of numerical errors (like the departure of translation invariance in grid codes due to errors in the reconstruction). ...