Chapter 11 polar coordinate Speaker LungSheng Chien Book

Chapter 11 polar coordinate Speaker: Lung-Sheng Chien Book: Lloyd N. Trefethen, Spectral Methods in MATLAB

Discretization on unit disk Consider eigen-value problem on unit disk with boundary condition We adopt polar coordinate , then , on Usually we take periodic Fourier grid in and , and non-periodic Chebyshev grid in 1 Chebyshev grid in Observation: nodes are clustered near origin Chebyshev grid in , for time evolution problem, we need smaller time-step to maintain numerical stability. 2 1 to 2 mapping

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 1 Eigenvalue of 2 Large eigenmode of is negative (real number) and does not approximate to Since ppw is too small such that resolution is not enough Mode N is spurious and localized near boundaries

Grid distribution 1 2

Preliminary: Chebyshev node and diff. matrix [1] Consider Chebyshev node on for Uniform division in arc Even case: Odd case:

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 for with identity Second derivative matrix is where is expressed as

Preliminary: Chebyshev node and diff. matrix [3] Let be the unique polynomial of degree define We abbreviate and with for , then impose B. C. In order to keep solvability, we neglect Similarly, we also modify differential matrix as , that is, zero

Preliminary: DFT Given a set of data point with [1] is even, Then DFT formula for for Definition: band-limit interpolant of If we write Also derivative is according to , then , is periodic sinc function

Preliminary: DFT Direct computation of derivative of Example: is a Toeplitz matrix. Second derivative is [2] , we have

Preliminary: DFT [3] For second derivative operation second diff. matrix is explicitly defined by using Toeplitz matrix (command in MATLAB)

Fornberg’s idea : extend radius to negative image [1] and 1 (odd): to avoid singularity of coordinate transformation 2 (even): to keep symmetry condition coordinate

Fornberg’s idea : extend radius to negative image [2] In general is odd, and Active variable is even, then and , total number is coordinate

Redundancy in coordinate transformation [1] 2 to 1 mapping coordinate redundant coordinate

Redundancy in coordinate transformation [2] 2 to 1 mapping coordinate redundant

Redundancy in coordinate transformation [3] is expressed as is odd, then Chebyshev differential matrix 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. 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

Redundancy in coordinate transformation [4] Symmetry property of Chebyshev differential matrix : -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 -2 2 -0. 1708 -0. 618 0. 8944 -1. 6180 is symmetric is NOT symmetric Permute column by -1. 1708 -2 -0. 618 0. 8944 2 -0. 1708 0. 8944 -1. 6180 -0. 8944 1. 618 -2 0. 1708 0. 6180 -0. 8944 1. 1708 2 is faster ?

Redundancy in coordinate transformation [5] is expressed as is odd, then Chebyshev differential matrix 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 -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 5. 7889 -10. 0669 7. 3167 2. 2111 -3. 6944 12. 6833 -31. 5331 neglect

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 5. 7889 -10. 0669 7. 3167 2. 2111 -3. 6944 12. 6833 -31. 5331 12. 6833 7. 3167 -10. 0669 2. 2111 -3. 6944 -1. 9056 5. 7889 is NOT sym. Permute column by -31. 5331 12. 6833 2. 2111 -3. 6944 7. 3167 -10. 0669 -1. 9056 5. 7889 7. 3167 -10. 0669 2. 2111 -3. 6944 -31. 5331 12. 6833 is NOT sym.

Row-major indexing: remove redundancy Define active variable for [1] and total number of active variables is 1 NOT 2 3 4 5 6 Index order 7 redundant 8 9 10 11 Index order 12

Row-major indexing: remove redundancy is odd, and suppose , then since is even, and Hence for and [2]

Row-major indexing: remove redundancy From symmetry condition, we have for and , symmetry condition implies Therefore, we have two important relationships 1 2 [3]

Kronecker product [1] 1 2 Define active variable for 3 4 and 5 6 Separation of variable: assume matrix A acts on r-dir and matrix B acts on is independent of Let is independent of be row-major index of active variable

Kronecker product is defined by Case 1: Case 2: [2]

Kronecker product Case 3: permutation, if permute [3]

Kronecker product Case 4: [4]

Non-active variable , on and is odd, and Active variable is Total number is Note that differential matrix , acts on is even, then , that is NOT is of dimension Neglect due to B. C. [1]

Non-active variable However and [2] 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

Non-active variable Recall for Consider Chebyshev differential matrix [3] and acts on and evaluate at We write in matrix notation Question: How about if we arrange equations on when fixed

Non-active variable [4] abbreviate We only keep operations on active variable That is, only consider equation Later on, we use the same symbol for

Non-active variable [5] Define permutation matrix Let Active variable with the same indexing in r-direction

Non-active variable Moreover, we modify differential matrix according to permutation P by where such that for and Evaluated at [6]

Non-active variable define and active variable To sum up [7]

Non-active variable Note that under row-major indexing, memory storage of is but If we adopt Kronecker products, then Definition: Kronecker product is defined by [8]

Non-active variable [9] The same reason holds for second derivative operator neglect equation on then collect equations on each we have where

Non-active variable We write second derivative operator on as where for so implies [10]

Non-active variable If we adopt Kronecker products, then [11]

Non-active variable summary 1 2 3 Note that so that all three system of equations are of the same order. [12]

Non-active variable , on and Discretization on then where [13]

Example: program 28 with boundary condition is odd and is even, let eigen-pair be sparse structure of Dimension :

Example: program 28 (mash plot of eigenvector) 1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue 2 Eigenvector is normalized by supremum norm,

Example: program 28 (nodal set)

Exercise 1 on annulus is odd and Chebyshev node on with boundary condition is even, let eigen-pair be Chebyshev node on

Exercise 1 (mash plot of eigenvector) 1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue 2 Eigenvector is normalized by supremum norm,