Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 130

notation that is widely used in the context of lattice Boltzmann the D indicates the
dimension and the Q the number of used velocities c i . The choice of the lattice has
an impact on several quantities like the speed of sound c s . Taking fewer velocities
into account decreases the computational costs but computational artifacts due to
the lattice nature become more severe, since certain directions might be favored,
violating Galilei invariance.
The collision operator represents inter-molecular interactions, i.e., collisions

between molecules that lead to a redistribution of f

Ω

. This redistribution
relaxes the single particle distribution function f towards a local equilibrium f eq ,
while it conserves mass and momentum. The actual shape of the collision operator
Ω

( x , c i , t )

can differ in the way how the equilibrium distribution f eq is calculated or how the
actual relaxation is realized. For a more detailed overview of the method the reader
is referred to the literature [ 19 , 28 , 29 , 50 , 51 , 79 , 80 ].
The code as it is used for the simulations in this chapter uses a BGK (or single
relaxation time) collision operator, similar to Eq. ( 16 ). The equilibrium distribution
function is given by

ρ f 1 u

9
2 (

3
2 u

f eq
i

2

( , )= + · + · ) − ·

x t w i 3 c i u c i u (18)

/ √ 3 is the speed

which is a polynomial expansion of the Maxwell distribution c s = 1

of sound for the D2Q9 lattice, u is the macroscopic velocity of the fluid.

Physical properties of the fluid are carried by the moments of the distribution f .
Namely the local density

( x , t )

∑ i

m 0 = ρ = f f i

the local momentum

∑ i

m 1 , 2 , 3 = j x , y , z = e x , y , z f i c i

and the momentum flux

m 3 ··· 18 = Π = ∑ i f i c i c i

The equilibrium distribution f eq is constructed such that it conserves local mass
and momentum but changes the viscous (or nonequilibrium) part of the momentum
flux

neq eq . It is realized by

Π = Π − Π

eq

= ∑ i

ρ f i (19)

j eq = ∑ i f i c i (20)

f eq
i

eq

= ∑ i Π c i c i = p 1 + ρ uu (21)