Quiz V, Math 572, 19 April, 2012
Name:
Solutions
1. (20 points): Consider the diusion problem
u(0, t) = 0 (t),
2u
u
=
t
x2
u(1, t) = 1 (t)
u(x, 0) = g (x)
for
0 <x<1,
for
for
0<tT ,
0tT ,
0x1.
(a) Write out the Crank-Nicolson approximation scheme for thi
Quiz IV, Math 572, 5 April, 2012
Name:
1. (10 points): Let m be a positive integer and set h = m1 . For any grid function
+1
w : cfw_0, 1, . . . , m, m + 1 R, with w0 = wm+1 = 0, prove that
m
m+1
i=1
2
w i w i 1
h
2
wi Ch
h
i=1
,
for some constant C > 0 t
Quiz III, Math 572, 8 March, 2012
Name:
Solutions
1. (10 points): Consider the BDF3 scheme
n+3
18
9
2
6
n+2 + n+1 n = hf (xn+3 , n+3 ).
11
11
11
11
Use the boundary locus method to argue that this is not an A-stable method.
Solution: According to the bou
Quiz II, Math 572, 23 February, 2012
Name:
1. (10 points): Show that the 2-step (implicit) Adams-Moulton method
i+2 i+1 = h
5
8
1
f (xi+2 , i+2 ) + f (xi+1 , i+1 ) f (xi , i )
12
12
12
is consistent, using the simple-to-check conditions (1) = 0 and (1) (1
Quiz I, Math 572, 9 February, 2012
Name:
Solutions
1. (10 points): Suppose that = limk xk and that
k
h2 1
| xk |
,
1 h2k
where > 0 and 0 < h < 1. Prove that the sequence cfw_xk converges to at least
quadratically.
Proof: Dene
k
h2 1
k
=
h2 .
k :=
2k
2k
Newtons Method and Variants
January 31, 2012
Denition 1. Suppose = limk xk . We say that the sequence cfw_xk converges to at least linearly
i there exists a sequence cfw_k of positive integers that converges to 0 and there exists (0, 1) such
that
k+1
|x
Midterm Exam, Math 572
15 March, 2012
Name:
Solutions
1. (10 points): Prove that the trapezoidal rule is A-stable.
Proof: The trapezoidal rule is
i+1 i =
h
[f (xi+1 , i+1 ) + f (xi , i )] .
2
Applying the scheme to the problem y = y , y (0) = 1, we have
i
Homework 13, Math 572
19 April, 2012
Name:
1. Consider the Crank-Nicolson scheme
n
+1
+1
w 1 2wn + wn+1 + wn1 2wn+1 + wn+1
wn+1 = wn +
2
for the diusion problem
u
2u
=
t
x2
u(0, t) = u(1, t) = 0
u(x, 0) = g (x)
=
s
,
h2
h=
1
,
m+1
w
0 <x<1,
for
for
0<tT
Homework 12, Math 572
12 April, 2012
Name:
1. Consider the scheme
wn+1 = wn + wn1 2wn + wn+1
bh n
w +1 wn1
2
for the convection-diusion problem
u
2u
u
=
b
2
t
x
x
u(0, t) = u(1, t) = 0
u(x, 0) = g (x)
for
0 <x<1,
for
for
0<tT ,
0tT ,
0x1,
where b > 0, =
Homework 11, Math 572
5 April, 2012
Name:
1. Consider the diusion problem
t u = xx u,
u(0, t) = 0 (t),
u(1, t) = 1 (t),
u(x, 0) = g (x),
0 < x < 1, 0 < t T
0 t T,
0 t T,
0 x 1,
T
where g (0) = 0 (0) and g (1) = 1 (0) for consistency. Dene h = m1 , s = N ,
Homework 10, Math 572
29 March, 2012
Name:
1. Let = (0, 1)2 and suppose Th is a triangulation of . Consider the piecewise
1
linear subspace of Sh H0 , dened as,
Sh = v C 0 () | v = 0 on , v |K P1 , K Th .
Let N be the number of interior nodes in the trian
Homework 9, Math 572
15 March, 2012
Name:
1. In the theory of the nite element method we need the Poincare inequality. Prove
the following version of it: for any u C [0, 1], u(0) = u(1) = 0,
1
1
2
u dx C
0
0
du
dx
2
dx,
for some constant C > 0 that does o
Homework 8, Math 572
8 March, 2012
Name:
1. Let m be a positive integer. Set h =
1
m+1
0
.
.
4
.
.
0,
1
4 1
0 1
4
4 1
1
Am = 0
.
.
.
0
and pk = kh. Dene Am Rmm via
0
.
.
1
and let Om , Im Rmm denote the zero and identity matrices, respectively. Now
2
2
Homework 7, Math 572
1 March, 2012
Name:
Updated 13 March, 2012
1. Use the boundary locus method to prove that BDF2 is A-stable.
2. If you have not already done so, prove that the BDF3 scheme
n+3
9
2
6
18
n+2 + n+1 n = hf (xn+3 , n+3 )
11
11
11
11
satise
Homework 6, Math 572
23 February, 2012
Name:
1. Prove that the linear stability domain for the trapezoidal rule, DTR , is equal to C .
2. Determine all the values of for which the -method,
n+1 = n + h f (xn , n ) + (1 )f (xn+1 , n+1 ) ,
is A-stable.
3. Sh
Homework 5, Math 572
16 February, 2012
Name:
1. The trapezoidal method for solving the IVP y (x) = f (x, y (x), x [a, b], y (x0 ) =
y0 , x0 [a, b), is dened as
i+1 = i +
h
[f (xi , i ) + f (xi+1 , i+1 )] ,
2
i = 0, 1, 2, . . . ,
with 0 = y0 , xi = x0 + ih
Homework 4, Math 572
9 February, 2012
Name:
1. Suppose that y (x) = f (x, y (x) on the interval [x0 , x1 ] with y (x0 ) = y0 . Assume
that a unique solution y exists such that it and all of its derivatives up to and
including the third order are dened and
Homework 3, Math 572
2 February, 2012
Name:
1. Find sequences that converge linearly, quadratically, and cubically, respectively.
2. Suppose that = limk xk and that
k
h2 1
,
1 h2k
where > 0 and 0 < h < 1. Prove that the sequence cfw_xk converges to at le
Homework 2, Math 572
26 January, 2012
Name:
Taylors Theorem: Let f C k [a, b], and suppose that f (k) is dierentiable on (a, b).
Then, for any x (a, b], there exists a = (x) (a, x) such that
f (x) = f (a) + f (a)(x a) + +
f (k) (a)
f (k+1) ( )
(x a)k +
(x
Homework 1, Math 572
19 January, 2012
Name:
1. Suppose that f C 2 (I ), where I is an interval containing a point , such that
f ( ) = 0, but f ( ) = 0 and f ( ) = 0. Prove that the sequence cfw_xk dened by
Newtons method,
f (xk )
,
xk+1 = xk
f ( xk )
co
Final Exam, Math 572
7 May, 2012
Name:
Solutions
1. (10 points): Consider the Upwind approximation scheme,
wn+1 = wn wn wn1 ,
=
as
,
h
for the Cauchy problem for the advection equation u + a u = 0, where a > 0.
t
x
Use the von Neumann stability analysis t
Class 28, Math 572
24 April, 2012
A Couple of Solved Problems
1. Consider the Crank-Nicolson scheme
wn+1 = wn +
n+1
+1
w 1 2wn+1 + wn+1 + wn1 wn + wn+1
2
2
for approximating the solution to the heat equation u = u on the intervals
t
x2
0 x 1 and 0 < t T