The first two methods can only be used if the body force is so small that
its effect is negligible, so
neither is suited to modelling a phenomena where gravity is important.
The two methods described in sections 5.1.4 and 5.1.5 give the
same equations of motion, however they are different.
The method described in
section 5.1.4 shall be called method (1) and the method described in
section 5.1.5 method (2).
The difference between the lattice Boltzmann operators
for models 1 and 2, acting on the same
distribution function 
is
The difference contains terms 
and 
.
During the implementation 
, the speed of sound and
. Thus, in general, the difference expressed in
equation (5.26) will be small. Although 
is small
it can still produce a significant effect. The difference
expressed in equation (5.26) being small does not imply
that 
is negligible in equation (5.3) or that
in equation (5.7) is negligible.
In section 5.1.4 the additional force affects the updated distribution
function, 
,
on each link, i=0,..7. In section 5.1.5 the updated distribution
function is altered, by the additional term, by an amount proportional
to 
. This means that 
is not changed (
will, however, be updated according to the standard collision operator).
If a
hexagonal grid, with links 2 and 5 horizontal, is used then neither
or 
will be changed by the additional term. If the grid is
orientated so that and are vertical then the additional term will
affect 
and it will have a larger effect on and
than it will on the others. A change in some of the terms at one
time-step will propagate through to all the terms at the
next time-step.
