Problems
299
8u
-+- 8 (U 2
-+gh ) =0,
and
Bhu
8 (2 h
2
- + - hu +g) =0,
8t
Bx
2
8h 8hu
- + - = 0 .
8t
8x
(a) Under what conditions do these systems have identical solutions?
(b) Give an example, including initial conditions and expressions for the
time-dependent solutions, for which these systems have different solutions.
(c) In those situations where these systems have different solutions, which
one serves as the correct mathematical model for shallow-water f1ow? (Hint:
The correct choice must be determined from fundamental physical principIes.)
2. Compute the speed at which the unit-arnplitude jump (5.3) must propagate
to be a weak solution to the conservation law
81/12 +!.- (21/13) = o.
8t
8x
3
How does this speed compare to that at which the same jump is propagated
by the inviscid Burgers's equation? Explain whether the difference in the
speed of these jumps is consistent with the sign of the inequality in the
entropy condition for solutions to Burgers's equation (5.12)?
3. Show that if 1/1(x, 0) 0, the solution to
-
81/1
+ -8 [c(x)1/I] = 0
8
8t
remains nonnegative for all t
x
o. Assurne that c and 1/1 have continuous derivatives in order to simplify the argument. (Hint: In order to develop negative 1/1, there must be a first time to and some point Xo for which
1/I(xo, to) = 0 and 1/1, (xo, to) < O. Show that this is impossible.) Does this
result generalize to problems in two and three spatial dimensions?
4. Use the results of Problem 3 to show that if
and 1/1 (x, 0) rp(x, 0), then 1/I(x, t) rp(x, t) for all x and t > O.
5. Suppose the constant-wind-speed advection equation (5.18) is approximated
using the Lax-Fredrichs scheme