§3.3 基本方程与技术

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.

All of them have to be closed by an equation of state, relating the gas pressure P to the internal energy (per unit mass) u and the density ρ. Assuming an ideal, monoatomic gas, this will be P = (γ−1) ρ u with the polytropic index γ = 5/3.

When applying these equations to the problem of cosmological structure formation, there are several features emerging in comparison to purely hydrodynamic simulations in a non-expanding background. First, the otherwise often neglected self-gravity, emerging as the ∇Φ term, has to be accounted for (its contribution can be solved following the methods described in N体章节).

Second, radiative losses, quantified through the cooling function Λ(u,ρ), where n²Λ (n²Λ/ρ) represents the cooling rate per unit volume (mass), with n being the gas number density, play a key role in influencing the evolution of the baryonic component (see also Section 11). Cooling is especially key to describe the history of star formation of the Universe, as outlined in Section 14.

Additionally, the equations have to be adapted to the cosmological background (to take into account the expansion history of the Universe). In almost all cases, the latter step is achieved by switching to the so called comoving coordinates and through a transformation of the time variable. This change does not typically alter the general structure of the equations above, except for additional, cosmological, time dependent pre-factors which appear in the equations (we refer the reader to chapter 5 in ^[50] for a rigorous derivation).