Pseudospectral Methods

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16 Νοε 2013 (πριν από 3 χρόνια και 1 μήνα)

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Pseudospectral Methods

Sahar

Sargheini

Laboratory of Electromagnetic Fields and Microwave Electronics (IFH)

ETHZ

1

7th Workshop on Numerical Methods for Optical Nano Structures, ETH Zurich, July 4
-
6, 2011

Pseudospectral Methods

2

Numerical methods

for solving PDEs


Approximate the

differential
operatore


Approximate the

solution

Finite difference

Spectral methods

)
(
)
(
L
x
f
x
u

f
u
a
a
a
x
R
N


L
)
,...,
,
,
(
2
1
0
)
,...,
,
(
)
(
U
1


dx
a
a
x
R
x
w
N
i



N
n
n
n
a
x
u
1
)
(

3

Pseudospectral Methods

Weighted residues

Galerkin

method

Least square

Pseudospectral


or


Collocation


or


method of selected points


i
i
w


)
(
i
i
x
x
w



i
i
w

L

Pseudospectral Methods


Finite Elements Method:

4

0
x
1
x
1

N
x
2

N
x
2
x
N
x
.
.
.
i
x
x
dx
du


Finite Difference Method:

0
x
1
x
1

N
x
2

N
x
2
x
N
x
.
.
.
i
x
x
dx
du

1


i
x
x
dx
du
1


i
x
x
dx
du

Pseudospectral

Methods

0
x
1
x
1

N
x
n
x
m
x
N
x
.
.
.
i
x
x
dx
du

.
.
.
.
.
.
Domain 1
Domain 2
Domain 3
N point
method

Pseudospectral Methods


Pseudospectral methods


Created by Kreiss and Oliger in 1972.


Were first introduced to the electromagnetic community around
1996 by Liu.

5

Error

)
(
N
h
O

]
)
/
1
[(
N
N
O

N
h
/
1

Infinite order / Exponential convergence

Memory usage and

time consumption

will be reduced

significantly

Pseudospectral Methods


Basis functions

6

Periodic functions

Trigonometric

Non periodic functions

Chebyshev or Legendre


Semi
-
Infinite functions

Laguerre

Infinite functions

Hermite

7

Fourier PSFD

Fourier PSFD

8



Liu extended the pseudospectral methods to the frequency domain (2002).



All proposed PSFD methods used Chebyshev basis functions.



However for periodic structures, trigonometric basis functions will be much more
suitable. In addition,using trigonometric functions, we can benefit from
characteristics of Fourier series, and that is why we call this method Fourier PSFD



Conventional single
-
domain PSFD methods suffer from staircasing error.



This error will not be reduced unless the number of discretization points increases.



To overcome this difficulty in a multidomain method, curved geometries should
be divided into several subdomains whereas this method is complicated and time
consuming to some extend.



We used a new technique to overcome the staircasing error in a single
-
domain
PSFD method.



We formulate the constitutive relations with the help of a convolution in the spatial
frequency domain.

Fourier PSFD


Constitutive relation

9

E
D
r


0



C
-
PSFD method



Conventional PSFD method

)
,
(
n
m
z
y
x
e
)
,
(
n
m
r
y
x

1
2
m
. . .
. . .
m+1
m
-
1
0
1
2
.
n
n
-
1
n+1
.
.
.
.
.


Conventional PSFD method














e
d
F
C
F
1
0


Bloch
-
Floquet:

r
k
j
e
r
h
r
H



)
(
)
(
r
k
j
e
r
e
r
E



)
(
)
(
Periodic functions



C
-
PSFD method

)
(
r
r

)
,
(
n
m
z
y
x
e
1
2
m
m+1
m
-
1
. . .
. . .
1
2
n
n
-
1
n+1
.
.
.
.
.
.
0
Fourier PSFD


Photonic crystals

a
r
08
.
0

a
r
4
.
0

10

a
r
4
.
0

1
1

r

9
.
8
2

r

TMz mods

C
-
PSFD: 6
×
6

Conventional PSFD: 6
×
6

Relative error of normalized frequency(%)
Number of discrete points in one direction (
N
)
C
-
PSFD
Conventional PSFD
5
10
15
20
25
30
35
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Relative error of normalized frequency(%)
Number of discrete points in one direction (
N
)
5
10
15
20
25
30
35
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Relative error of normalized frequency(%)
Number of discrete points in one direction (
N
)
100
2

r

9
.
8
2

r

TEz mods

r
X
M
r
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized frequency
Wave vector
PWE
C
-
PSFD
C
-
PSFD: 10
×
10

5
10
15
20
25
30
35
0
2
4
6
8
10
12
14
16
18
Relative error of normalized frequency(%)
Number of discrete points in one direction (
N
)
Error: Second band at


the M point of the first Brillouin zone

Fourier PSFD

11


Photonic crystals

4
/
1
a
x


4
/
1
a
y

5
/
1
a
r

4
/
2
a
x

6
/
2
a
y


7
/
2
a
r

1
1

r

9
.
8
2

r

TMz modes

C
-
PSFD
Conventional PSFD
Relative error of normalized frequency(%)
Number of discrete points in one direction (
N
)
Error: Second band at


the M point of the first Brillouin zone

0
0.5
1


0
0
a
/

a
/

a
/


a
/


x
k
y
k
Normalized frequency
C
-
PSFD: 8
×
8

Fourier PSFD

12


Photonic crystals

a
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Wave vector
X
r
M
r
Normalized Frequency
TMz modes

C
-
PSFD: 12
×
12

5
10
15
20
25
30
35
0
5
10
15
20
25
Number of discrete points in one direction (N)
Relative error of normalized frequency (%)
Error: seventeenth band at

a
k
x
/


a
k
y
/
2
.
0


1
1

r

4
.
11
2

r

Fourier PSFD


13


Left
-
handed binary grating

i
E
d
L
1
d
2
d
Left handed material
)
,
(
1
1


)
,
(
2
2


y
z
1

2

1
R
2
R
3
R
)
,
(
1
1


)
,
(
2
2









n
z
y
n
r
z
jk
y
n
L
k
j
C
x
z
y
E
n
)
)
2
(
exp(
ˆ
)
,
(








n
z
y
n
t
z
jk
y
n
L
k
j
D
x
z
y
E
n
)
)
2
(
exp(
ˆ
)
,
(












n
n
z
j
n
z
j
n
n
p
n
n
e
B
e
A
y
E
x
z
y
E


)
(
ˆ
)
,
(
L
)
,
(
1
1


)
,
(
2
2


z
y
0
d


)
1
,
1
(
,
1
1

r
r




)
2
,
2
(
,
2
2



r
r


2
/
L
d

0

y
k
-1
-0.5
0
0.5
1
-25
-20
-15
-10
-5
0
5
10
15
20
25
)
/
(
Real
0
k


3
.
1

L
0
d
L
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
y
Amplitude
6322
.
0
0546
.
1
/
0
j
k



Fourier PSFD


4
.
1

L
14


Left
-
handed binary grating

y
z


)
1
,
1
(
,
1
1

r
r




)
2
,
2
(
,
2
2



r
r


2
/
L
d

0

y
k
L
d

1
0
0
d1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
z
Amplitude

9345
.
0

y
Thank you for your attention

15