In the remainder of this section, we illustrate the material balance approach for developing finite difference equations. Discussion of the Taylor series approach for obtaining FDEs is left to the exercises.
To formulate a FDE for a one dimensional conservation law, we begin by
establishing a grid of points on the xt-plane with step size h in the x
direction and step size k in the
t-direction.
Let spatial grid points be denoted by 
and time grid points be
, where n and j are integers and 
is the origin
of the space--time grid.
See Figure 1.
Figure 1: Representative grid points and a control interval.
In Figure 1, the points 
and 
are introduced to
establish a ``control interval''.
A development of a conservation law FDE proceeds along the same lines as
presented in the previous section.
We begin with a conservation statement similar to Eq. 1.
Equation 10 states that the change in material content of the
interval
from
time 
to time 
is given by the flow of material M
into this interval at 
minus
the flow out of the interval at 
from time 
to time 
.
Equation 10 is an integral conservation law
equation that exactly expresses
the conservation of material M in the case that
no sources or sinks are present.
It is the basis for our development of a variety of conservation law finite
difference equations.
To obtain a FDE that is useful in a computational setting requires that the physical variables, density r and flux q, be related to the field variable u. For simplicity, we begin with the case in which density is assumed to have the form
with c and b constants.
