## Abstract

After reading this chapter, you will know the basics of the lattice Boltzmann method, how it can be used to simulate fluids, and how to implement it in code. You will have insight into the derivation of the lattice Boltzmann equation, having seen how the continuous Boltzmann equation is discretised in velocity space through Hermite series expansion, before being discretised in physical space and time through the method of characteristics. In particular, you will be familiar with the various simple sets of velocity vectors that are available, and how the discrete BGK collision model is applied.

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## Notes

- 1.
Since the Boltzmann equation is more general, it also has solutions that

*do not*correspond to Navier-Stokes solutions. The connections between these two equations will be further explored in Sect. 4.1 - 2.
There is a limit to how small a velocity set can be, though, as it has to obey the requirements shown in (3.60) to be suitable for Navier-Stokes simulations. The smallest velocity set in 3D is D3Q13, but it has the disadvantage of being tricky to apply.

- 3.
This statement is not proven in this chapter, but rather in Sect. 12.1 on sound waves.

- 4.
Usually, the velocity sets \(\{{\boldsymbol c}_{i}\}\) are chosen such that any spatial vector \({\boldsymbol c}_{i}\varDelta t\) points from one lattice site to a neighbouring lattice site. This guarantees that the populations

*f*_{ i }always reach another lattice site during a time step*Δ t*, rather than being trapped between them. - 5.
Other collision operators are available which use additional relaxation times to achieve increased accuracy and stability (cf. Chap. 10).

- 6.
As a consequence of this, mass and momentum are conserved in collisions. This conservation can be expressed mathematically as

*∑*_{ i }*Ω*_{ i }= 0 and \(\sum _{i}{\boldsymbol c}_{i}\varOmega _{i} ={\boldsymbol 0}\). - 7.
- 8.
For example, for a moderate simulation domain of 100 × 100 × 100 lattice sites and the D3Q19 velocity set, one copy of the populations

*f*_{ i }requires about 145 MiB of memory, where we assume double precision (8 bytes per variable). - 9.
As we will see later in Chap. 10, the collision operator can have different forms all of which locally conserve the moments (mass, momentum and energy) and, thus, yield the correct macroscopic behaviour.

- 10.
This is the case for sufficiently smooth functions.

- 11.
The goal of this book is to give practical aspects of the derivation and usage of the LBM rather than a rigorous mathematical theory for some assumptions used. However, for interested readers we recommend [7] for a rigorous proof.

- 12.
In the isothermal case, only density and momentum are considered.

- 13.
Another example of the higher versatility of the D3Q27 velocity set is its ability to reproduce the Navier-Stokes dynamics with higher-order Galilean invariance, through an off-lattice implementation [21].

- 14.
- 15.
We have described the general principles of such finite volume methods in Sect. 2.1.2.

- 16.
Applying the trapezoidal rule when integrating along characteristics is not in general the best approach. As a simple example, applying the trapezoidal rule to the equation d

*f*∕d*ζ*= −*f*^{3}gives us*f*(*ζ*_{0}+*Δ ζ*) −*f*(*ζ*_{0}) = −[*f*(*ζ*_{0})^{3}+*f*(*ζ*_{0}+*Δ ζ*)^{3}]*Δ t*∕2, which would have to be solved implicitly, instead of the exact explicit result \(f(\zeta _{0}+\varDelta \zeta ) = 1/\sqrt{2\varDelta t + 1/f(\zeta _{0 } )^{2}}\). A more general second-order LB discretisation is shown in Appendix A.5, though in this case the end result is the same as here. - 17.
Note that is more convenient for code implementation to write (3.83) in the form

$$\displaystyle{ f_{i}^{\star }({\boldsymbol x},t) = \left (1 -\frac{\varDelta t} {\tau } \right )f_{i}({\boldsymbol x},t) + \frac{\varDelta t} {\tau } f_{i}^{\mathrm{eq}}({\boldsymbol x},t). }$$The specific choice

*τ*=*Δ t*(which is quite common in LB simulations) leads to the extremely efficient collision rule$$\displaystyle{ f_{i}^{\star }({\boldsymbol x},t) = f_{ i}^{\mathrm{eq}}({\boldsymbol x},t), }$$i.e. the populations directly go to their equilibrium and forget about their previous state. We provide more details about efficient implementations of (3.83) in Chap. 13

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Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M. (2017). The Lattice Boltzmann Equation. In: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-44649-3_3

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