CME 306
Numerical Solutions to Partial Differential Equations
http://stanford.edu/
~
pgarapon/cme306.html
Part I  Finite difference techniques
1. Prove that the scheme above is consistent with equation (
??
) Using
Taylor expansions, one shows:
u
n
+1
j

2
u
n
j
+
u
n

1
j
(Δ
t
)
2
=
∂
2
u
∂t
2
+
O
(Δ
t
2
)
δ
2
(
δ
2
(
u
)) =
h
4
∂
4
u
∂x
4
+
O
(
h
6
)
Which gives the consistency.
2. Rewrite the scheme without the term
σ
. One substitutes and gets:
u
n
+1
j

2
u
n
j
+
u
n

1
j
(Δ
t
)
2
+
ν
u
n
j

2

4
u
n
j

1
+ 6
u
n
j

4
u
n
j
+1
+
u
n
j
+2
(
h
)
4
= 0
3. Write the matrix
D
in
M
N
(
R
) (matrix of size
N
, with real entries)
corresponding to
δ
2
. Write the scheme in a matrix form and explicitly define
the matrices involved. Denoting (
U
n
)
j
=
u
n
j
, one gets the vector equality:
U
n
+1
= (2
I
+
cD
2
)
U
n

U
n

1
where
c
=
ν
Δ
t
2
h
4
and
D
is the matrix:
D
=
2

1
0
...
...
1

1
2

1
...
...
..
0

1
2

1
...
..
...
...
...
...
...
...
..
...
...

1
2

1
1
..
...
..

1
2
4. Following the Von Neumann approach, write a relation between the in
tensities of the Fourier mode
k
at times
n

1,
n
and
n
+1: ˆ
u
n
+1
(
k
)
,
ˆ
u
n
(
k
)
,
ˆ
u
n

1
(
k
).
Classic Fourier transform approach yields the recurrence:
(after a little
trigonometry)
ˆ
u
n
+1
(
k
) =

ˆ
u
n

1
(
k
) + (2

16
c
sin
4
(
kπh
))ˆ
u
n
(
k
)
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
which can be written in matrix form:
ˆ
u
n
+1
(
k
)
ˆ
u
n
(
k
)
=
2

16
c
sin
4
(
kπh
)

1
1
0
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 KEN
 H0, ... ..., Hilbert space, variational formulation

Click to edit the document details