3.5 Matrix construction
The construction of each matrix equation involves building the matrix and source coefficients from the terms in the equation being solved, with further adjustments for boundary conditions.
The figure below illustrates the process of
building a matrix equation for a field 
from an equation
including advection, diffusion and source 
.
and source 
are calculated for each individual term in the
equation, e.g. 
, 
, etc. using the discretisation methods
described in this chapter.
The overall coefficients are calculated as the sum of coefficients for each term in the the equation. Most terms contribute to both the matrix and source coefficients, although this depends on the choice of discretisation scheme.
Finally, boundary conditions are incorporated
into the equation through further adjustments to coefficients in
and 
as shown below. The adjustments, principally from the
advection and diffusion terms, are applied to coefficients
corresponding to cells at the domain boundary.
Implicit and explicit
The equation for a field 
on the previous page
is discretised to form the matrix equation 
. The discretisation of
a term is implicit when it
contributes to coefficients in 
by treating
as
the solved field 
.
Explicit
discretisation calculates coefficients in 
only, using current
values of fields. When solving an equation for 
, derivatives without
must be explicit. Terms with 
could be treated
explicitly by using current values of 
, but they generally
are not, since explicit solutions are unstable beyond a limiting
time step as described in Sec. 3.17
. A notable exception to this are
the terms discussed in Sec. 3.20
.
The curl derivative, e.g. 
, includes terms in
and 
in the decoupled matrix equation for 
. These terms must be
treated explicitly since they do not include 
itself. The situation
for 
and 
is the same, so the curl derivative can be explicit
only.
This leaves the following terms which are generally treated implicitly:3
;
;
; 
, where 
is a scalar.This chapter details: the discretisation of
these terms, which are generally treated implicitly; and, other
terms such as divergence 
and gradient 
which can only be
discretised explicitly within a segregated solution.
