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.