After application of the space discretization of momentum and transport equations and the implementation of the boundary conditions as described in Section 7, the discretized equations in the time-domain read:
where 
and 
denote algebraic vectors containing
the velocity and pressure unknowns in grid points, 
is the
discrete scalar grid function and the total number
of scalar unknowns is given by N. For two-equation turbulence models in the
absence of other physical effects, N = 2. Furthermore, M is the diagonal
matrix containing the value of 
in the centroids on the diagonal, D and
G are the discretized divergence
and gradient operators, S represents the space discretization of the convection
and viscous stress tensors and 
is an operator involving the discretization
of convection and diffusion of the scalar. In fact
this term may also be non-linear, but in our program it is treated as if it is
linear. The
vector 
contains the volume forces and boundary values of the
velocities and 
represents the source term with respect
to , which is generally a function of and
, 
and the boundary conditions.
The extra source term 
result from
the anti-diffusive parts as
deferred corrections to the first order upwind approximation.
The time discretization is performed with a standard technique for the
solution of ordinary differential equations. At this moment only one type
of time-solver is present: the so-called 
method, i.e. a linear
combination of the forward and backward Euler schemes.
Generalized method ??????? (Jos, Guus)
