**1. Introduction**

In the kinetic theory, the distribution function of a rarefied gaseous system is governed by the Boltzmann equation or its models [1]. In the applications, the discretization of these equations in the velocity (and physical) space is usually performed. One of the most popular discretization approaches is the Lattice–Boltzmann (LB) method [2–5] which was initially developed as an alternative to the continuum fluid methods like Navier– Stokes equations [6]; furthermore, the method has been extended to the rarefied flows modeling [7–19]. The conventional LB model has the following form

$$\frac{df\_i}{dt} = \frac{1}{\pi} (f\_i^{eq} - f\_i)\_{\prime} \quad i = 1 \dots N\_{\prime}$$

where *fi*(*t*, *x*) is the distribution function related to the particles with the velocity *ci*, *i* = 1 ... *N*, *τ* is the relaxation time, *f eq <sup>i</sup>* is the local equilibrium, *N* is the number of the discrete velocities, *<sup>d</sup> dt* <sup>=</sup> *<sup>∂</sup> <sup>∂</sup>t*<sup>+</sup> *ci <sup>∂</sup> <sup>∂</sup>r*, *<sup>r</sup>* is the spatial variable. In this approach, the collisions between the particles are described in a phenomenological way, i.e., it is postulated that, due to the collisions, the distribution function tends to the local equilibrium state at a rate proportional to *f eq <sup>i</sup>* − *fi*. For LB models, the local equilibrium is usually taken as a finite-order polynomial on the bulk velocity, and the conservation laws for mass and momentum are satisfied by construction. On the other hand, for this form of the local equilibrium, the *H*-theorem does not exist [20–22]. To overcome this issue, models with non-polynomial equilibria have been proposed [23–26].

Another possible discretization technique is the discrete velocity (DV) Boltzmann method [27–30], the general DV Boltzmann model reads as

$$\frac{df\_i}{dt} = \sum\_{jkl}^{N} A\_{kl}^{ij} (f\_k f\_l - f\_i f\_j) \equiv I\_i[f\_1, \dots, f\_N], \quad i = 1 \dots N,\tag{1}$$

where *Aij kl* <sup>=</sup> *<sup>A</sup>kl ij* ≥ 0 are the transition probabilities.

**Citation:** Ilyin, O. Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics. *Mathematics* **2021**, *9*, 993. https:// doi.org/10.3390/math9090993

Academic Editors: Mikhail Posypkin, Andrey Gorshenin, Vladimir Titarev and Janos Sztrik

Received: 27 March 2021 Accepted: 20 April 2021 Published: 28 April 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Compared to the LB method, the DV models have some attractive properties. Similarly to the Boltzmann equation, the binary collisions are described explicitly. Moreover, by construction, the *H*-theorem is valid for these models [28], i.e.,

$$\frac{dH}{dt} \le 0, \quad H(t) = \int dr \sum\_{i}^{N} f\_i \log(f\_i).$$

Moreover, the local equilibrium for DV kinetic Boltzmann models can be obtained as an exponential function of the macroscopic variables. The DV Boltzmann approach attracted the attention of many researchers several decades ago, but, at present, is significantly less popular than the LB method. For instance, the well-known four velocity Broadwell equation in two dimensions has been investigated thoroughly [31–36], this model has correct collision invariants, but its discrete velocity set is too small and lacks isotropy [37]; therefore, the correct description of the hydrodynamics is impossible in the framework of this model. In addition, another subtle feature should be mentioned: for the discrete velocity models, the molecular chaos hypothesis can be violated, i.e., the particles can be correlated before the collision [38]. This is undesirable, but the influence of this effect on the flow properties in applications is not clear. Furthermore, one should construct the DV Boltzmann models in such a way that the only conserved variables are mass, momentum and energy. The equilibrium state is obtained as minimum of the *H*-function under the constraint that these variables are not changed by collisions. The presence of other conserved quantities (spurious invariants) changes the form of the local equilibrium state, this, in turn, leads to a distortion of the hydrodynamic equations. The construction of DV Boltzmann models without excessive invariants is a non-trivial procedure [39–42].

In this paper we consider a DV Boltzmann model on a nine velocity, two-dimensional lattice. As a starting point, we consider the local equilibrium for the general DV Boltzmann model and its expansion at the vicinity of the absolute Maxwell distribution. Next, the Chapman–Enskog expansion for the DV Boltzmann model is performed in order to derive the hydrodynamic equations. In addition, we show that the model does not have invariants without physical meaning. The considered model has four different possible transition probabilities. In terms of the LB theory, this model can be considered as a scheme with multiple relaxation times. For viscosity, we obtain a closed expression depending on the values of the transition probabilities. If the viscosity is fixed, we obtain a constraint on the transition probabilities, but three of them can be chosen as free parameters; for instance, they can be adjusted to improve stability properties. As benchmark problems, we consider the shear wave decay and Taylor–Green vortex. The numerical experiments show excellent agreement between the numerical simulation results and analytical solutions.
