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?
Explain step in Lattice Boltzmann method
lattice boltzmann methods