Boundary Condition
The network of points that define the positive and negative characteristics in ℜ 4 can
now be determined. Before we proceed, however, it is necessary to recall one of our
previous equations, namely, u ¼ (R + + R À )/2; hence, the boundary condition, u ¼ 0,
at the end wall gives the requirement that
R À ¼ ÀR þ ,
hence, the negative characteristic reflected off the end wall carries the same magnitude as the Riemann invariant of the incident positive characteristic but bearing a
negative sign, and this will be used for calculating the network of points in ℜ 4 .
Calculating the Coordinates of a Boundary Point
Let us now determine the coordinates of P 1 at the boundary wall as shown in
Fig. A.6. The slope of the positive characteristic evaluated at b with coordinates
(9.32, 8.877) and that joins the points b and P 1 is
dt
dx
C þ
¼
5
3R þ þ 2R À
¼
5
3 Â 5:75
ð
Þþ 2 Â À6
ð
Þ
¼ 0:952
and the slope of the positive characteristic evaluated at P 1 that joins the points b and
P 1 is
dt
dx
C þ
¼
5
3R þ þ 2R À
¼
5
3 Â 5:75
ð
Þþ 2 Â À5:75
ð
Þ
¼ 0:869
where the previously referred to boundary condition has been applied. The average
of these slopes is 0.911, and by using the relationship,
dt
dx
C þ
¼
Δt
Δx
we have
t 1 À 8:877
10 À 9:32
¼ 0:911
for the time t 1 at P 1 , hence, t 1 ¼ 9.496 and the coordinates of P 1 are (10, 9.496).
Other boundary points are calculated in a similar manner.
Appendix A
325