MPO 662 { Problem Set 3
1. Consider the Lax Fredrichs approximation to the scalar advection equation:
u
n
+1
j
?
u
n
j
+1
+
u
n
j
?
1
2
t
+
c
u
n
j
+1
?
u
n
j
?
1
2
x
= 0
Determine the truncation error for this scheme
Study the consistency of this FD representation
What condition to you have to impose on
x
so that the time errors and spatial
errors are of the same order of magnitude?
Determine the stability characteristics of this scheme.
2. The 1D heat equation,
u
t
=
u
xx
, is a parabolic equation. It is discretized using an
implicit scheme of the form:
u
n
+1
j
?
u
n
j
t
=
1
2
"
u
n
+1
j
+1
?
2
u
n
+1
j
+
u
n
+1
j
?
1
x
2
+
u
n
j
+1
?
2
u
n
j
+
u
n
j
?
1
x
2
#
(1)
Study the stability and convergence of this method for periodic b oundary conditions.
Hint: expand in Taylor series about (
j
x;
(
n
+
1
2
)
t
), do the space-expansion rst, and
use it in analyzing the time-series.
3. Write a program to solve numerically the equation:
h
t
+ (
hu
)
x
+ (
hv
)
y
= 0 (2)
u
=
?
2
y
?
1
2
; v
= 2
x
?
1
2
;
(3)
h
(
x; y ; t
= 0) = max(0
;
1
?
8
r
)
; r
2
=
x
?
1
4
2
+
y
?
1
2
2
(4)
in the unit square 0
x; y
1. Use a conservative scheme discretized on a C-grid and
the following uxes:
a Donor cell ux
a 2
nd
order centered ux
Choose a time-stepping scheme that is at least as accurate as the space discretization
(you can reuse some of the programs you wrote earlier). Contour the numerical solution
versus the analytical solution at time
t
= 1 for a max. Courant numbers of
0
:
25 and
compute the root mean square error and the maximum error in the solution. Monitor
the minimum and maximum values of the solution throughout the calculation, as well
as the total budget of
h
; check to see if
h
is indeed conserved. Use 4 sets of grids using
16
16, 32
32, 64
64, and 128
128 cells and discuss the numerical results. Draw
