The most commonly used lattice Boltzmann models are summarized on this page. Be aware though that opinions sometimes differ on how certain models should be classified. The following is a rough guideline which must be taken with a certain flexibility.
The lattice Boltzmann model with BGK collision operator, or BGK model for short, is the classic in LB fluid models. Historically, the success of the LB method is to a large extent founded on BGK. This model is most often used to solve the incompressible Navier-Stokes equations. In that case, it uses a quasi-compressible approach, in which the fluid is contrived into adopting a slightly compressible behavior to solve the pressure equation. The BGK method can also be used to simulate compressible flows at low Mach-number, but both the bulk viscosity and the speed of sound are lattice constants which can not be adjusted (there exist workarounds for both limitations though). The BGK model has been extended in numerous attempts, such as those listed on this page, to enhance numerical stability and/or accuracy for specific problems, or to represent additional physical phenomena. It is however observed that while these extensions improve some aspects of BGK, they also introduce new problems. BGK stays therefore the model of choice in many situations, because of its ease of implementation as well as its reliability. A good review of the BGK model is found in Chen and Doolen 1998.
The BGK collision operator acts on the off-equilibrium part of all particle populations on a lattice node, multiplying all of them with the same relaxation parameter. An extension of this, described by the concept of a General Collision Matrix, is to act with a linear operator on a vector consisting of all off-equilibrium particle populations. In the case that this operator is diagonalizable, its eigenvalues possess a nice interpretation as relaxation parameters acting on modes of the particle populations. Some modes are physically meaningful, and tackling with them modifies the physics of the model. But some of them (the ghost modes) are not, and they can be played with to enhance the numerical stability of the model. Technically speaking any model with a diagonalizable General Collision Matrix can be viewed as a Multiple-Relaxation-Time model, i.e., a model in which the relaxation parameter varies from a mode to another. However, when people speak about MRT, they often refer to the model referenced in D'Humières e.a. 2002, in which the relaxation parameters of the ghost modes are obtained from a linear stability analysis.
In the regularized model, better accuracy and stability are obtained by eliminating higher order, non-hydrodynamic terms from the particle populations. This model is based on the observation that the hydrodynamic limit of the BGK model is not dependent on the details of the particle populations, but only on the value of the first three moments (density, velocity and stress tensor). The idea is to compute these three moments at every time step and on each node. The particle populations are then “regularized”, i.e. they are assigned a new value depending only on these moments. After this, the usual BGK collision is executed. Executing consecutively a regularization and a BGK collision has the effect of a diagonalizable linear operation on the off-equilibrium part of the particle populations. This model can therefore be attributed to the family of Multiple-Relaxation-Time models. The value of the relaxation parameters is however determined from a physical argument, based on the Chapman-Enskog expansion of BGK, and not on a numerical stability analysis. The regularized model is introduced in Latt and Chopard 2006.
The entropic lattice Boltzmann (ELB) model is similar to the BGK one. The collision operator is again a simple relaxation towards an equilibrium. The main differences are the evaluation of the equilibrium distribution function and a local modification of the relaxation time. The equilibrium is no more taken as a discretization of the continuous Maxwell-Boltzmann equilibrium distribution but rather as an extremum of the discretized entropy under the conservations constrains of the system (mass and momentum usually). The relaxation time is tuned locally in order to avoid an entropy decrease during collision. This last procedure is shown to prevent the distribution functions from adopting negative values, which ensures unconditional numerical stability. Unfortunately, it also has a high computational cost since one has to solve an implicit equation at each lattice node for each time step. A description of this model can be found in Ansumali 2002.