Problems
171
I 0,
- kn + 1)2 if Ix - kl s k;
otherwise.
and use the initial eondition
1fr(x,O) =
(a) Given that
1
2/ 3 dx
-
1/3 c(x)
= 1.391,
find the eorreet x-Ioeation of the peak of the initial distribution at time t =
3. Deseribe the shape and loeation of the true solution at t = 3.
(b) Compare numerieal solutions obtained using the seeond-order approximations
(3.129)
and
+
with the fourth -order spaee sehemes
= 0
and
+ 3" 4( (c) x }X - 3" 1( (c) 2x }2X = O.
Assurne that c and rP are loeated at the same points. Use
= 1/32 and
a Courant number of 0.6 based on the maximum wind speed. Take a single forward step to initialize the leapfrog integration. Plot the left of the
domain and show the solutions at t = 0, 1.5, and 3. Also plot the wind
speed .
(e) Retry the above simulations using
= 1/64 and diseuss the degree of
improvement.
(d) Now try adding a fourth-order spatial filter to eaeh scheme in the
1/32 ease. Lag the filter in time . For example , (3.129) beeomes
=
02t'l'j + C02x'I'j = -Y
»»
A.n
(A.n-I
'l'j+2 -
AA.n-1
""f'j+1 +
6A.n-1
'l'j
-
AA.n-1
""f'j_1 +
A.n-I)
'l'j-2 .
Diseuss the dependenee of the solution on the parameter Y
As astart,
set Y
= 0.01.