Chapter 11 polar coordinate

1
Chapter 11 polar coordinate

• Speaker Lung-Sheng Chien

Book Lloyd N. Trefethen, Spectral Methods in
MATLAB
2
Discretization on unit disk
Consider eigen-value problem on unit disk
with boundary condition
We adopt polar coordinate
, then
, on
and
, and non-periodic Chebyshev grid in
Usually we take periodic Fourier grid in
1
Chebyshev grid in
Chebyshev grid in
Observation nodes are clustered near origin
, for time evolution problem,
we need smaller time-step to maintain numerical
stability.
2
1 to 2 mapping
3
Asymptotic behavior of spectrum of Chebyshev
diff. matrix
In chapter 10, we have showed that spectrum of
Chebyshev differential matrix (second order)
approximates
with eigenmode
Eigenvalue of
is negative (real number) and
1
Large eigenmode of
does not approximate to
2
Since ppw is too small such that resolution is
not enough
Mode N is spurious and localized near boundaries
4
Grid distribution
1
2
5
Preliminary Chebyshev node and diff. matrix 1
Consider
Chebyshev node on
for
Uniform division in arc
Even case
Odd case
6
Preliminary Chebyshev node and diff. matrix 2
Given
Chebyshev nodes
and corresponding function value
We can construct a unique polynomial of degree
, called
is a basis.
where differential matrix
is expressed as
for
for
where
with identity
Second derivative matrix is
7
Preliminary Chebyshev node and diff. matrix 3
and
Let
be the unique polynomial of degree
with
define
and
for
, then impose B.C.
,that is,
We abbreviate
In order to keep solvability, we neglect
,that is,
zero
neglect
zero
neglect
Similarly, we also modify differential matrix as
8
Preliminary DFT 1
Given a set of data point
with
is even,
Then DFT formula for
for
for
Definition band-limit interpolant of
, is periodic sinc function
If we write
, then
Also derivative is according to
9
Preliminary DFT 2
, we have
Direct computation of derivative of
Example
is a Toeplitz matrix.
Second derivative is
10
Preliminary DFT 3
For second derivative operation
second diff. matrix is explicitly defined by
using Toeplitz matrix (command in MATLAB)
11
Fornbergs idea extend radius to negative image
1
and
(odd) to avoid singularity of coordinate
transformation
1
(even) to keep symmetry condition
2
coordinate
coordinate
12
Fornbergs idea extend radius to negative image
2
In general
is odd, and
is even, then
and
Active variable
, total number is
coordinate
coordinate
13
Redundancy in coordinate transformation 1
2 to 1 mapping
coordinate
coordinate
redundant
14
Redundancy in coordinate transformation 2
2 to 1 mapping
coordinate
coordinate
redundant
15
Redundancy in coordinate transformation 3
is odd, then Chebyshev differential matrix
is expressed as
8.5
-10.4721
2.8944
-1.5279
1.1056
-0.5
2.6180
-1.1708
-2
0.8944
-0.618
0.2764
-0.7236
2
-0.1708
-1.6180
0.8944
-0.382
0.3820
-0.8944
1.618
0.1708
-2
0.7236
-0.2764
0.6180
-0.8944
2
1.1708
-2.618
0.5
-1.1056
1.5279
-2.8944
10.4721
-8.5
-1.1708
-2
0.8944
-0.618
2
-0.1708
-1.6180
0.8944
-0.8944
1.618
0.1708
-2
0.6180
-0.8944
2
1.1708
neglect
16
Redundancy in coordinate transformation 4
Symmetry property of Chebyshev differential
matrix
-1.1708
-2
0.8944
-0.618
-1.1708
-2
is symmetric
2
-0.1708
-1.6180
0.8944
2
-0.1708
-0.8944
1.618
0.1708
-2
0.8944
-0.618
0.6180
-0.8944
2
1.1708
is NOT symmetric
-1.6180
0.8944
Permute column by
-1.1708
-2
0.8944
-0.618
2
-0.1708
-1.6180
0.8944
is faster ?
-0.8944
1.618
0.1708
-2
0.6180
-0.8944
2
1.1708
17
Redundancy in coordinate transformation 5
is odd, then Chebyshev differential matrix
is expressed as
41.6
-68.3607
40.8276
-23.6393
17.5724
-8
21.2859
-31.5331
12.6833
-3.6944
2.2111
-0.9528
-1.8472
7.3167
-10.0669
5.7889
-1.9056
0.7141
0.7141
-1.9056
5.7889
-10.0669
7.3167
-1.8472
-0.9528
2.2111
-3.6944
12.6833
-31.5331
21.2859
-8
17.5724
-23.6393
40.8276
-68.3607
41.6
-31.5331
12.6833
-3.6944
2.2111
7.3167
-10.0669
5.7889
-1.9056
-1.9056
5.7889
-10.0669
7.3167
2.2111
-3.6944
12.6833
-31.5331
neglect
18
Redundancy in coordinate transformation 6
Symmetry property of Chebyshev differential
matrix
-31.5331
12.6833
-3.6944
2.2111
7.3167
-10.0669
5.7889
-1.9056
-1.9056
5.7889
-10.0669
7.3167
-31.5331
12.6833
2.2111
-3.6944
12.6833
-31.5331
is NOT sym.
7.3167
-10.0669
Permute column by
-3.6944
2.2111
is NOT sym.
5.7889
-1.9056
-31.5331
12.6833
-3.6944
2.2111
7.3167
-10.0669
5.7889
-1.9056
-1.9056
5.7889
-10.0669
7.3167
2.2111
-3.6944
12.6833
-31.5331
19
Row-major indexing remove redundancy 1
Define active variable
for
and
total number of active variables is
1
2
NOT
3
4
5
6
Index order
7
8
redundant
9
10
11
12
Index order
20
Row-major indexing remove redundancy 2
is odd, and
, then
since
suppose
is even, and
Hence for
and
21
Row-major indexing remove redundancy 3
From symmetry condition, we have
for
and
, symmetry condition implies
Therefore, we have two important relationships
1
2
22
Kronecker product 1
1
2
Define active variable
3
4
for
and
5
6
Separation of variable assume matrix A acts on
r-dir and matrix B acts on
is independent of
Let
be row-major
index of active variable
is independent of
23
Kronecker product 2
Kronecker product is defined by
Case 1
Case 2
24
Kronecker product 3
Case 3 permutation, if permute
25
Kronecker product 4
Case 4
26
Non-active variable ? active variable 1
, on
and
is odd, and
is even, then
Active variable is
, that is
Total number is
NOT
Note that differential matrix
is of dimension
,acts on
Neglect due to B.C.
27
Non-active variable ? active variable 2
However
and
act on
NOT active variable, how to deal with?
From previous discussion, we have following
relationships which can solve this problem
and
where
is a permutation matrix
28
Non-active variable ? active variable 3
Recall
for
and
Consider Chebyshev differential matrix
acts on
and evaluate at
We write in matrix notation
Question How about if we arrange equations on
when fixed
29
Non-active variable ? active variable 4
abbreviate
We only keep operations on active variable
That is, only consider equation
for
Later on, we use the same symbol
30
Non-active variable ? active variable 5
Define permutation matrix
Let
Active variable with the same indexing in
r-direction
31
Non-active variable ? active variable 6
Moreover, we modify differential matrix according
to permutation P by
where
such that for
and
Evaluated at
32
Non-active variable ? active variable 7
define
and active variable
To sum up
33
Non-active variable ? active variable 8
Note that under row-major indexing, memory
storage of
is
but
If we adopt Kronecker products, then
Definition Kronecker product is defined by
34
Non-active variable ? active variable 9
The same reason holds for second derivative
operator
neglect equation on
then
collect equations on each
we have
where
35
Non-active variable ? active variable 10
We write second derivative operator on
as
where
for
implies
so
36
Non-active variable ? active variable 11
If we adopt Kronecker products, then
37
Non-active variable ? active variable 12
summary
1
2
3
Note that
so that all three system of equations are of the
same order.
38
Non-active variable ? active variable 13
, on
and
Discretization on
then
where
39
Example program 28
with boundary condition
is even, let eigen-pair be
is odd and
sparse structure of
Dimension
40
Example program 28 (mash plot of eigenvector)
Eigenvalue is sorted, monotone increasing and
normalized to first eigenvalue
1
Eigenvector is normalized by supremum norm,
2
41
Example program 28 (nodal set)
42
Exercise 1
with boundary condition
on annulus
is even, let eigen-pair be
is odd and
Chebyshev node on
Chebyshev node on
43
Exercise 1 (mash plot of eigenvector)
Eigenvalue is sorted, monotone increasing and
normalized to first eigenvalue
1
Eigenvector is normalized by supremum norm,
2