A collision model was proposed by Bhatnagar, Gross and Krook  (BGK) to simplify the analysis of the lattice Boltzmann equation. Using LB-BGK approximation equation (1) can be written as
Luo, "Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation," Physical Review, vol.
The Lattice Boltzmann equation (LBE) can be determined as
The combination of streaming and collision steps in a 9-velocity square lattice modified the lattice Boltzmann equation as
This method couples the flexible structure (e.g., capsule) dynamics and the fluid dynamics by using the immersed boundary method and calculates the incompressible viscous power-law fluid motion by solving the lattice Boltzmann equation
. In order to achieve the non-Newtonian rheology, a shear rate-dependant relaxation time is employed.
Lin, "A stable discretization of the lattice Boltzmann equation
for simulation of incompressible twophase flows at high density ratio," Journal of Computational Physics, vol.
Viscous flow computations with the method of lattice Boltzmann equations
. Progress in Aerospace Sciences.
Dellar, "Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations
," Physical Review E--Statistical, Nonlinear, and Soft Matter Physics, vol.
Chen, "Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation
," Physical Review E, vol.
Succi, The Lattice Boltzmann Equation
for Fluid Dynamics and Beyond, Oxford University Press, Oxford, UK, 2001.
The Lattice Boltzmann equation
 (LBE) is linear, but actually its nonlinearity is embedded in the left side of the LBE.