$c_s$	speed of sound	$\xi$	lattice velocity
$f$	density distribution function	$w$	weight coefficient of the lattice
$\mathbf{f}$	boundary force terms of the Euler point	$\rho$	denisity
$\mathbf{F}$	boundary force terms of the Lagrangian point	$\upsilon$	kinematic viscosity
$K$	kernel function	$\mu$	dynamic viscosity
$Kn$	Knudsen number	$\tau$	relaxation time
$M$	number of the Euler points
$N$	number of the Lagrangian point	Superscripts
$\mathbf{r}$	position of the Euler point	$’$	perturbation part
$s$	index of the Lagrangian point	$\overline{}$	steady mean part
$\mathbf{u}$	velocity vector	$eq$	equilibrium distribution function
${\mathbf{U}}_I$	velocity of the fluid	$neq$	non-equilibrium distribution function
${\mathbf{U}}_B$	velocity of the immersed boundary	$*$	intermediate variables
$\mathbf{R}$	position of the Lagrangian point	$n+1$	variables at the next time step
${\delta }_t$	time step
${\delta }_x$	lattice spacing	Subscripts
${\delta }_e$	scale of the Euler grid	$\alpha$	direction
${\delta }_l$	scale of the Lagrangian grid	$e$	Euler grid point
$\lambda$	wavelength	$l$	Lagrangian grid point