The paper presents a detailed review of the smooth particle hydrodynamics (SPH) method with particular focus on its astrophysical applications. We start by introducing the basic ideas and concepts and thereby outline all ingredients that are necessary for a practical implementation of the method in a working SPH code. Much of SPH's success relies on its excellent conservation properties and therefore the numerical conservation of physical invariants receives much attention throughout this review. The self-consistent derivation of the SPH equations from the Lagrangian of an ideal fluid is the common theme of the remainder of the text. We derive a modern, Newtonian SPH formulation from the Lagrangian of an ideal fluid. It accounts for changes of the local resolution lengths which result in corrective, so-called "grad-h-terms". We extend this strategy to special relativity for which we derive the corresponding grad-h equation set. The variational approach is further applied to a general-relativistic fluid evolving in a fixed, curved background space-time. Particular care is taken to explicitely derive all relevant equations in a coherent way.
Deep Dive into Astrophysical Smooth Particle Hydrodynamics.
The paper presents a detailed review of the smooth particle hydrodynamics (SPH) method with particular focus on its astrophysical applications. We start by introducing the basic ideas and concepts and thereby outline all ingredients that are necessary for a practical implementation of the method in a working SPH code. Much of SPH’s success relies on its excellent conservation properties and therefore the numerical conservation of physical invariants receives much attention throughout this review. The self-consistent derivation of the SPH equations from the Lagrangian of an ideal fluid is the common theme of the remainder of the text. We derive a modern, Newtonian SPH formulation from the Lagrangian of an ideal fluid. It accounts for changes of the local resolution lengths which result in corrective, so-called “grad-h-terms”. We extend this strategy to special relativity for which we derive the corresponding grad-h equation set. The variational approach is further applied to a general-r
Much of what we observe in the physical Universe has been shaped by fluid dynamical processes. From the hot gas in galaxy clusters to the internal structures of their consituent galaxies down to their stars and planets, all has been formed by the interplay between gravity, gas dynamics and further physical processes such as the interaction with radiation, nuclear burning or magnetic fields. The latter processes often involve intrinsic length and time scales that are dramatically different from those of the gas dynamical processes, therefore many astrophysical problems are prime examples of multi-scale and multiphysics challenges. The complexity of the involved physical processes and the lack of symmetry usually prohibit analytical treatments and only numerical approaches are feasible. Although fluid dynamics is also crucial for many technical applications, their requirements usually differ substantially from those of astrophysics and this also enters the design of the numerical methods. Typical astrophysical requirements include:
• Since fixed boundaries are usually absent, flow geometries are determined by the interplay between different physical processes such as gas dynamics and (self-)gravity which often lead to complicated, dynamically changing flow geometries. Thus, a high spatial adaptivity is often required from astrophysical hydodynamics schemes. • Shocks often crucially determine the evolution of cosmic objects. Examples include in supernova remnants or the Earth’s magnetosphere. • Physical quantities can vary by many orders of magnitudes between different regions of the simulation domain. This requires a particularly high robustness of the numerical scheme. • In many astrophysical problems the numerical conservation of physically conserved quantities determines the success and the reliability of a computer simulation. Consider, for example, a molecular gas cloud that collapses under the influence of its own gravity to form stars. If the simulation for some reason dissipates angular momentum, a collapsing, self-gravitating portion of gas may form just a single stellar object instead of a multiple system of stars and it will thus produce a qualitatively wrong result. • Many astrophysical questions require dealing with physical processes beyond gas dynamics and self-gravity. A physically intuitive and flexible formulation of the numerics can substantially facilitate the implementation of new physics modules into existing codes.
No numerical method performs equally well at each of the above requirements, therefore, the choice of the best-suited numerical approach can often save a tremendous amount of effort in obtaining reliable results. Therefore: horses for courses.
In the following, the smooth particle hydrodynamics (SPH) method [1][2][3][4][5][6], a completely mesh-free approach to solve the hydrodynamic equations is discussed in detail. Its conservation properties are a major strength of SPH, therefore “hard-wired” conservation receives much attention throughout this text. The derivation of the SPH equations (in the absence of dissipation) requires nothing more than a suitable Lagrangian, a density prescription that depends on the coordinates and the first law of thermodynamics. The resulting equations conserve the physically conserved quantities even in their discretized form, provided that the original Lagrangian possessed the correct symmetries. Therefore, derivations from Lagrangians play a central role in our discussion of the subject. This review has emerged from a lecture series on “Computational relativistic astrophysics” as part of a Doctoral Training Programme on the “Physics of Compact Stars” that was held in summer 2007 at the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT * ) in Trento, Italy. In the pedagogical spirit of this lecture series the text is kept in “lecture” rather than “paper” style, i.e. even rather trivial steps are written down explicitely. The goal is to pave a broad and smooth avenue to a deep understanding of the smooth particle hydrodynamics method rather than just to provide a bumpy trail. Due to this pedagogical scope, the focus of this review needs to be clear-cut, but rather narrow: it only discusses the numerical solution of the inviscid hydrodynamics equations, in the Newtonian, specialrelativistic and general-relativistic (fixed metric) case.
For practical astrophysical simulations often sophisticated additional physics modules are required and their implementation may pose additional numerical challenges which are beyond the scope of this review. In the following we provide a brief list of additional physics that has been implemented into SPH and point to references that are intended as starting points for further reading.
• Gravity: Self-gravity is for many astrophysical problems a key ingredient. A straight forward calculation of pairwise gravitational forces between N particles requires a prohibitively large O(N
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