The positive characteristic emanating from x i and the negative characteristic
emanating from x i + 1 intersect at the point (x i , x i + 1 ). Since the Riemann invariants
remain constant along the characteristics we can determine the quantities u i, i + 1 and
c i, i + 1 at the first intersection points from the equations,
u i,iþ1 þ
2c i,iþ1
γ À 1
¼ R
i
ð Þ
þ
ð2:100Þ
u i,iþ1 À
2c i,iþ1
γ À 1
¼ R
iþ1
ð
Þ
À
:
ð2:101Þ
where the Riemann invariants R
i
ð Þ
þ and R
iþ1
ð
Þ
þ
are given by Eqs. (2.98) and (2.99).
By adding and subtracting these latter two equations, we obtain,
u i,iþ1 ¼
1
2
R
i
ð Þ
þ þ R
iþ1
ð
Þ
À
ð2:102Þ
and
c i,iþ1 ¼
γ À 1
ð
Þ
4
R
i
ð Þ
þ À R
iþ1
ð
Þ
À
,
ð2:103Þ
hence, u i, i + 1 and c i, i + 1 are now known at these first intersection points. Having
determined the values of u and c at the intersection points, we must now find the
location of these points and we know, in general, that the characteristics are not
straight lines; hence, an approximate method must be applied. A good approximation
1 for the coordinates of the intersection points can be obtained by averaging the
slope of the positive characteristic at (x i , 0) and (x i, i + 1 , t i, i + 1 ), and, similarly by
averaging the slope of the negative characteristic at (x i + 1 , 0) and (x i, i + 1 , t i, i + 1 ).
Hence, the average slope of the positive characteristic is
dx
dt
( )
C þ
¼
1
2
u i þ c i
ð
Þþ u i,iþ1 þ c i,iþ1
ð
Þ
½
Š
ð 2:104Þ
and the average slope of the negative characteristic is
dx
dt
( )
C À
¼
1
2
u iþ1 À c iþ1
ð
Þþ u i,iþ1 À c i,iþ1
ð
Þ
½
Š ,
ð2:105Þ
hence, we can write
1 The approximation will improve if the spacing between the initial set of data points is reduced.
86
2 Waves of Finite Amplitude