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8
if x0 < 0
< x0 + t
x = : x0 + (1 ; x0 )t if 0 x < 1 :
(10.1.19)
x
if 1 x
0 Several characteristics are shown in Figure 10.1.7. The characteristics rst intersect at
t = 1. After that, the solution would presumably be multivalued, as shown in Figure
10.1.8.
It's, of course, quite possible for multivalued solutions to exist however, (i) they
are not observed in physical situations and (ii) they do not satisfy (10.1.11a) in any
classical sense. Discontinuous solutions are often observed in nature once characteristics
of the corresponding conservation law model have intersected. They also do not satisfy 12 Hyperbolic Problems
t 1 1 x Figure 10.1.7: Characteristics for Burgers' equation (10.1.16) with initial data given by
(10.1.18).
u(x,0) 0 u(x,1/2) 1 2 x 0 u(x,1) 0 1 2 x 2 x u(x,3/2) 1 2 x 0 1 Figure 10.1.8: Multivalued solution of Burgers' equation (10.1.16) with initial data given
by (10.1.18). The solution u(x t) is shown as a function of x for t = 0, 1/2, 1, and 3/2.
(10.1.11a), but they might satisfy the integral form of the conservation law (10.1.1). We
examine the simplest case when two classical solutions satisfying (10.1.11a) are separated
by a single smooth curve x = (t) across which u(x t) is discontinuous. For each t > 0
we assume that < (t) < and let superscripts  and + denote conditions immediately 10.1. Conservation Laws 13 to the left and right, respectively, of x = (t). Then, using (10.1.1), we have
d Z udx = d Z ; udx + Z udx] = ;f (u)j
dt
dt
+
or, di erentiating the integrals
Z;
Z
; _; +
ut dx + u
ut dx ; u+ _+ = ;f (u)j :
+ The solution on either side of the discontinuity was assumed to be smooth, so (10.1.11a)
holds in ( ;) and ( + ) and can be used to replace the integrals. Additionally, since
is smooth, _; = _+ = _. Thus, we have ;f (u)j ; + u; _ ; f (u)j ; u
+ + _ = ;f (u)j or
_(u+ ; u;) = f (u+) ; f (u;): (10.1.20) q] q+ ; q; (10.1.21a) Let
denote the jump in a quantity q and write (10.1.20) as u] _ = f (u)]: (10.1.21b) Eq...
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 Spring '14
 JosephE.Flaherty

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