# Explain step in Lattice Boltzmann method

Actually I am reading a book about the Lattice Boltzmann methods, and here is a quotation:

After introducing BGKW approximation, the Boltzmann equation (without external forces) can be approximated as, $$\dfrac{\partial f}{\partial t}+c\cdot \nabla f=\dfrac1\tau\left(f^{eq}-f\right)\tag{1}$$ In lattice Boltzmann method, the above equation is descretized and assumed it is valid along specific diresctions, linkages. Hence, the discrete Boltzmann equation can be written along a specified direction as, $$\dfrac{\partial f_i}{\partial t}+c_i \nabla f_i=\dfrac1\tau\left(f_i^{eq}-f_i\right)\tag{2}$$

I cannot understand how can I pass from equation $(1)$ to $(2$) (notice also the absence of the dot product between $c_i$ and $\nabla f_i$) can you please explain? is it a typo?

• Yes, I think there is a typo. From what I understand of LBM, the approximation consists in "freezing" single directions; usually the coordinate directions $x_i$. This would correspond to replacing $c$ in (1) by the vector $c_i e_i$, i.e., the vector where all but the $i$th component are set to zero. If this is the case, then in (2) it should actually be $c_i\partial_i f_i$, where $\partial_i$ is the partial derivative in direction $x_i$ and $f_i$ is just to denote that this is the approximation of $i$ corresponding to the direction $x_i$. – Christian Clason Jul 1 '16 at 20:59
• @ChristianClason: Can you please share with some books on lattice boltzmann methods, the most books are very advanced, is there any easy to read booksfor engineers? – s.ouchene Jul 1 '16 at 21:12
• I'm sorry, I don't know any. I know very little about LBM myself. (But some things just are advanced, and there's no way around it. Sometimes you have to read two books!) – Christian Clason Jul 1 '16 at 22:19
• @Navaro - you should check out my answer here for a list of references. Specifically, Sukop's book is good as an initial introduction. Unfortunately, it will not help with answering your question which is more suited for a book like Succi's. – nluigi Jul 4 '16 at 15:43
• @Navaro - it is indeed a typo; $c$ is a continuous particle velocity, while $c_i$ is a discretized particle velocity. It is still a velocity however and as such is a vector. The above equation should therefor be written as: $\partial_{t}f_{i}+c_{i}\cdot\nabla f_{i}=-\frac{\left\​ (f_{i}-f_{i}^{eq}\right)}{\tau}$‌​ Note, that the discretized velocity is a constant vector (it's mapped to a lattice) and therefor it is allowed to do this: $c_i\cdot\nabla f_i = \nabla \cdot c_if_i$ – nluigi Aug 16 '16 at 12:20

In (1), $f$ is an unknown function of $t$, $x$ and $\xi$. Here $\xi$ denotes the moment of a particle. Note that (1) is only a scalar equation for $f=f(t,x,\xi)$.
From (1) to (2), we actually discretize the space of $\xi$, from continuous space $-\infty<\xi<\infty$ to discretize space $\xi_1$, $\xi_2$, $\dots$, $\xi_n$. Now (2) is a set of $n$ equations for functions $f_i=f_i(t,x)$, $i=1,\dots,n$.
• $f$ is a function of time $t$, position $\mathbf r$, and velocity $\mathbf c$ – s.ouchene Jul 2 '16 at 1:13