In this section we restrict ourselves to the time discretization of the momentum
equations. The treatment of the transport equations will be done in exactly the
same manner. Furthermore, for brevity the
arguments 
will be
dropped from the operator S. Application of the 
-method to
(8.1), (8.2) gives
where lies between zero and unity, n denotes the preceding time level,
n+1 the new time level and 
is the time step. For 
and
we obtain the first order explicit and implicit Euler schemes,
respectively, and for 
we have the second order Crank-Nicolson
scheme. The -method is unconditionally stable for 
.
In the range 
a time-step restriction is necessary.
At this moment 
has not been tested, except .
To solve (8.4), (8.5) it is necessary to linearize the
term 
. In ISNaS, the convective terms are linearized
by a Newton linearization as given in formula (4.7). Coefficients
that depend on the solution, like for example the viscosity, are evaluated
at the preceding time-level.
Practical implementation:
Instead of solving (8.4), (8.5) immediately, we introduce an
intermediate level 
by:
If we assume that 
is linearized, i.e. can be written as
, then
(8.4) reduces to:
Substitution of (8.6) into (8.7) and (8.5) gives
From (8.6) it then follows that:
Once the momentum equations have been solved for 
, each of the transport
equations for 
is solved for one time step. Exactly the same
methodology as described above is employed. Quantities already computed, like the
velocity are substituted in these equations, thus improving the stability.
However, it should be noted that the source term
contains deferred corrections which are evaluated
explicitly, so that no contribution to coefficients of the system to be
solved for the new time level is involved.
